problem solving in algebraic equations

Solving Equations

What is an equation.

An equation says that two things are equal. It will have an equals sign "=" like this:

That equations says:

what is on the left (x − 2)  equals  what is on the right (4)

So an equation is like a statement " this equals that "

What is a Solution?

A Solution is a value we can put in place of a variable (such as x ) that makes the equation true .

Example: x − 2 = 4

When we put 6 in place of x we get:

which is true

So x = 6 is a solution.

How about other values for x ?

• For x=5 we get "5−2=4" which is not true , so x=5 is not a solution .
• For x=9 we get "9−2=4" which is not true , so x=9 is not a solution .

In this case x = 6 is the only solution.

You might like to practice solving some animated equations .

More Than One Solution

There can be more than one solution.

Example: (x−3)(x−2) = 0

When x is 3 we get:

(3−3)(3−2) = 0 × 1 = 0

And when x is 2 we get:

(2−3)(2−2) = (−1) × 0 = 0

which is also true

So the solutions are:

x = 3 , or x = 2

When we gather all solutions together it is called a Solution Set

The above solution set is: {2, 3}

Solutions Everywhere!

Some equations are true for all allowed values and are then called Identities

Example: sin(−θ) = −sin(θ) is one of the Trigonometric Identities

Let's try θ = 30°:

sin(−30°) = −0.5 and

−sin(30°) = −0.5

So it is true for θ = 30°

Let's try θ = 90°:

sin(−90°) = −1 and

−sin(90°) = −1

So it is also true for θ = 90°

Is it true for all values of θ ? Try some values for yourself!

How to Solve an Equation

There is no "one perfect way" to solve all equations.

A Useful Goal

But we often get success when our goal is to end up with:

x = something

In other words, we want to move everything except "x" (or whatever name the variable has) over to the right hand side.

Example: Solve 3x−6 = 9

Now we have x = something ,

and a short calculation reveals that x = 5

Like a Puzzle

In fact, solving an equation is just like solving a puzzle. And like puzzles, there are things we can (and cannot) do.

Here are some things we can do:

• Add or Subtract the same value from both sides
• Clear out any fractions by Multiplying every term by the bottom parts
• Divide every term by the same nonzero value
• Combine Like Terms
• Expanding (the opposite of factoring) may also help
• Recognizing a pattern, such as the difference of squares
• Sometimes we can apply a function to both sides (e.g. square both sides)

Example: Solve √(x/2) = 3

And the more "tricks" and techniques you learn the better you will get.

Special Equations

There are special ways of solving some types of equations. Learn how to ...

• solve Quadratic Equations
• solve Radical Equations
• solve Equations with Sine, Cosine and Tangent

Check Your Solutions

You should always check that your "solution" really is a solution.

How To Check

Take the solution(s) and put them in the original equation to see if they really work.

Example: solve for x:

2x x − 3 + 3 = 6 x − 3     (x≠3)

We have said x≠3 to avoid a division by zero.

Let's multiply through by (x − 3) :

2x + 3(x−3) = 6

Bring the 6 to the left:

2x + 3(x−3) − 6 = 0

Expand and solve:

2x + 3x − 9 − 6 = 0

5x − 15 = 0

5(x − 3) = 0

Which can be solved by having x=3

Let us check x=3 using the original question:

2 × 3 3 − 3 + 3  =   6 3 − 3

Hang On: 3 − 3 = 0 That means dividing by Zero!

And anyway, we said at the top that x≠3 , so ...

x = 3 does not actually work, and so:

There is No Solution!

That was interesting ... we thought we had found a solution, but when we looked back at the question we found it wasn't allowed!

This gives us a moral lesson:

"Solving" only gives us possible solutions, they need to be checked!

• Note down where an expression is not defined (due to a division by zero, the square root of a negative number, or some other reason)
• Show all the steps , so it can be checked later (by you or someone else)

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How to Solve an Algebraic Expression

Last Updated: April 6, 2024 Fact Checked

This article was co-authored by David Jia . David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math. There are 10 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 491,614 times.

An algebraic expression is a mathematical phrase that contains numbers and/or variables. Though it cannot be solved because it does not contain an equals sign (=), it can be simplified. You can, however, solve algebraic equations , which contain algebraic expressions separated by an equals sign. If you want to know how to master this mathematical concept, then see Step 1 to get started.

Understanding the Basics

• Algebraic expression : 4x + 2
• Algebraic equation : 4x + 2 = 100

• 3x 2 + 5 + 4x 3 - x 2 + 2x 3 + 9 =
• 3x 2 - x 2 + 4x 3 + 2x 3 + 5 + 9 =
• 2x 2 + 6x 3 + 14

• You can see that each coefficient can be divisible by 3. Just "factor out" the number 3 by dividing each term by 3 to get your simplified equation.
• 3x/3 + 15/3 = 9x/3 + 30/3 =
• x + 5 = 3x + 10

• (3 + 5) 2 x 10 + 4
• First, follow P, the operation in the parentheses:
• = (8) 2 x 10 + 4
• Then, follow E, the operation of the exponent:
• = 64 x 10 + 4
• Next, do multiplication:
• And last, do addition:

• 5x + 15 = 65 =
• 5x/5 + 15/5 = 65/5 =
• x + 3 = 13 =

Joseph Meyer

To solve an equation for a variable like "x," you need to manipulate the equation to isolate x. Use techniques like the distributive property, combining like terms, factoring, adding or subtracting the same number, and multiplying or dividing by the same non-zero number to isolate "x" and find the answer.

Solve an Algebraic Equation

• 4x + 16 = 25 -3x =
• 4x = 25 -16 - 3x
• 4x + 3x = 25 -16 =
• 7x/7 = 9/7 =

• First, subtract 12 from both sides.
• 2x 2 + 12 -12 = 44 -12 =
• Next, divide both sides by 2.
• 2x 2 /2 = 32/2 =
• Solve by taking the square root of both sides, since that will turn x 2 into x.
• √x 2 = √16 =
• State both answers:x = 4, -4

• First, cross multiply to get rid of the fraction. You have to multiply the numerator of one fraction by the denominator of the other.
• (x + 3) x 3 = 2 x 6 =
• Now, combine like terms. Combine the constant terms, 9 and 12, by subtracting 9 from both sides.
• 3x + 9 - 9 = 12 - 9 =
• Isolate the variable, x, by dividing both sides by 3 and you've got your answer.
• 3x/3 = 3/3 =

• First, move everything that isn't under the radical sign to the other side of the equation:
• √(2x+9) = 5
• Then, square both sides to remove the radical:
• (√(2x+9)) 2 = 5 2 =
• Now, solve the equation as you normally would by combining the constants and isolating the variable:
• 2x = 25 - 9 =

• |4x +2| - 6 = 8 =
• |4x +2| = 8 + 6 =
• |4x +2| = 14 =
• 4x + 2 = 14 =
• Now, solve again by flipping the sign of the term on the other side of the equation after you've isolated the absolute value:
• 4x + 2 = -14
• 4x = -14 -2
• 4x/4 = -16/4 =
• Now, just state both answers: x = -4, 3

Community Q&A

• The degree of a polynomial is the highest power within the terms. Thanks Helpful 9 Not Helpful 1
• Once you're done, replace the variable with the answer, and solve the sum to see if it makes sense. If it does, then, congratulations! You just solved an algebraic equation! Thanks Helpful 7 Not Helpful 3
• To cross-check your answer, visit wolfram-alpha.com. They give the answer and often the two steps. Thanks Helpful 8 Not Helpful 5

You Might Also Like

• ↑ https://www.math4texas.org/Page/527
• ↑ https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-expressions-and-variables/cc-6th-combining-like-terms/v/combining-like-terms-2
• ↑ https://www.mathsisfun.com/algebra/factoring.html
• ↑ https://www.mathsisfun.com/operation-order-pemdas.html
• ↑ https://sciencing.com/tips-for-solving-algebraic-equations-13712207.html
• ↑ https://www.mathsisfun.com/algebra/equations-solving.html
• ↑ https://tutorial.math.lamar.edu/Classes/Alg/SolveExpEqns.aspx
• ↑ https://www.mathsisfun.com/algebra/fractions-algebra.html
• ↑ https://math.libretexts.org/Courses/Coastline_College/Math_C045%3A_Beginning_and_Intermediate_Algebra_(Chau_Duc_Tran)/10%3A_Roots_and_Radicals/10.07%3A_Solve_Radical_Equations
• ↑ https://www.mathplanet.com/education/algebra-1/linear-inequalitites/solving-absolute-value-equations-and-inequalities

About This Article

If you want to solve an algebraic expression, first understand that expressions, unlike equations, are mathematical phrase that can contain numbers and/or variables but cannot be solved. For example, 4x + 2 is an expression. To reduce the expression, combine like terms, for example everything with the same variable. After you've done that, factor numbers by finding the lowest common denominator. Then, use the order of operations, which is known by the acronym PEMDAS, to reduce or solve the problem. To learn how to solve algebraic equations, keep scrolling! Did this summary help you? Yes No

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Algebraic Equations

Algebraic equations are two algebraic expressions that are joined together using an equal to ( = ) sign. An algebraic equation is also known as a polynomial equation because both sides of the equal sign contain polynomials. An algebraic equation is built up of variables, coefficients, constants as well as algebraic operations such as addition, subtraction, multiplication, division, exponentiation, etc.

If there is a number or a set of numbers that satisfy the algebraic equation then they are known as the roots or the solutions of that equation. In this article, we will learn more about algebraic equations, their types, examples, and how to solve algebraic equations.

What is Algebraic Equations?

An algebraic equation is a mathematical statement that contains two equated algebraic expressions. The general form of an algebraic equation is P = 0 or P = Q, where P and Q are polynomials . Algebraic equations that contain only one variable are known as univariate equations and those which contain more than one variable are known as multivariate equations. An algebraic equation will always be balanced. This means that the right-hand side of the equation will be equal to the left-hand side.

Algebraic Expressions

A polynomial expression that contains variables, coefficients, and constants joined together using operations such as addition , subtraction, multiplication, division, and non-negative exponentiation is known as an algebraic expression . An algebraic expression should not be confused with an algebraic equation. When two algebraic expressions are merged together using an "equal to" sign then they form an algebraic equation. Thus, 5x + 1 is an expression while 5x + 1 = 0 will be an equation.

Algebraic Equations Examples

x 2 - 5x = 3 is a univariate algebraic equation while y 2 x - 5z = 3x is an example of a multivariate algebraic equation.

Types of Algebraic Equations

Algebraic equations can be classified into different types based on the degree of the equation. The degree can be defined as the highest exponent of a variable in an algebraic equation. Suppose there is an equation given by x 4 + y 3 = 3 5 then the degree will be 4. In determining the degree, the exponent of the constant or coefficient is not considered. The number of roots of an algebraic equation depends on its degree. An algebraic equation where the degree equals 5 will have a maximum of 5 roots. The various types of algebraic equations are as follows:

Linear Algebraic Equations

A linear algebraic equation is one in which the degree of the polynomial is 1. The general form of a linear equation is given as a 1 x 1 +a 2 x 2 +...+a n x n = 0 where at least one coefficient is a non-zero number. These linear equations are used to represent and solve linear programming problems.

Example: 3x + 5 = 5 is a linear equation in one variable . y = 2x - 6 is a linear equation in two variables .

Quadratic Algebraic Equations

An equation where the degree of the polynomial is 2 is known as a quadratic algebraic equation . The general form of such an equation is ax 2 + bx + c = 0, where a is not equal to 0.

Example: 3x 2 + 2x - 6 = 0 is a quadratic algebraic equation. This type of equation will have a maximum of two solutions.

Cubic Algebraic Equations

An algebraic equation where the degree equals 3 will be classified as a cubic algebraic equation . ax 3 + bx 2 + cx + d = 0 is the general form of a cubic algebraic equation (a ≠ 0).

Example: x 3 + x 2 - x - 1 = 0. A cubic algebraic equation will have a maximum of three roots as the degree is 3.

Higher-Order Polynomial Algebraic Equations

Algebraic equations that have a degree greater than 3 are known as higher-order polynomial algebraic equations. Quartic (degree = 4), quintic (5), sextic (6), septic (7) equations all fall under the category of higher algebraic equations. Such equations might not be solvable using a finite number of operations.

Algebraic Equations Formulas

Algebraic equations can be simplified using several formulas and identities. These help to expedite the process of solving a given equation. Given below are some important algebraic formulas :

• (a + b) 2 = a 2 + 2ab + b 2
• (a - b) 2 = a 2 - 2ab + b 2
• (a + b)(a - b) = a 2 - b 2
• (x + a)(x + b) = x 2 + x(a + b) + ab
• (a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3
• (a - b) 3 = a 3 - 3a 2 b + 3ab 2 - b 3
• a 3 + b 3 = (a + b)(a 2 - ab + b 2 )
• a 3 - b 3 = (a - b)(a 2  + ab + b 2 )
• (a + b + c) 2 = a 2 + b 2 + c 2 + 2ab + 2bc + 2ca
• Quadratic Formula : [-b ± √(b² - 4ac)]/2a
• Discriminant : b 2 - 4ac

How to Solve Algebraic Equations

There are many different methods that are available for solving algebraic equations depending upon the degree. If an algebraic equation has two variables then two equations will be required to find the solution. Thus, it can be said that the number of equations required to solve an algebraic equation will be equal to the number of variables present in the equation. Given below are the ways to solve algebraic equations.

A linear algebraic equation in one variable can be solved by simply applying basic arithmetic operations  on both sides of the equation.

E.g: 4x + 1 = 5.

4x = 5 - 1 (Subtracted 1 from both sides).

4x = 4 (Solve the R.H.S using algebraic operations)

x = 1 (Divided both sides by 4)

Linear algebraic equations in more than one variable will be solved using the concept of simultaneous equations .

A quadratic algebraic equation can be solved by using identities , factorizing , long division, splitting the middle term, completing the square , applying the quadratic formula, and using graphs . A quadratic equation will always have a maximum of two roots.

E.g: x 2 + 2x + 1 = 0

Using the identity (a + b) 2 = a 2 + 2ab + b 2 , we get

a = x and b = 1

(x + 1) 2 = 0

(x + 1)(x + 1) = 0

x = -1, -1.

The most effective way of solving higher-order algebraic polynomials in one variable is by using the long division method. This decomposes the higher-order polynomial into polynomials of a lower degree thus, making it easier to find the solutions.

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Important Notes on Algebraic Equations:

• An algebraic equation is an equation where two algebraic expressions are joined together using an equal sign .
• Polynomial equations are algebra equations.
• Algebraic equations can be one-step, two-step , or multi-step equations .
• Algebra equations are classified as linear, quadratic, cubic, and higher-order equations based on the degree.
• Example 1: Solve the algebraic equation x + 3 = 2x Solution: Taking the variable terms on one side of the equation and keeping the constant terms on the other side we get, 3 = 2x - x 3 = x Answer: x = 3
• Example 2: A total of 15 items can fit in a box. If the box contains 2 scales, 7 pencils, and 1 eraser then how many pens can fit in the box? Solution: Converting this problem statement in the form of an algebraic equation we get, 2 scales + 7 pencils + 1 eraser + x pens = 15 2 + 7 + 1 + x = 15 Solving the L.H.S 10 + x = 15 x = 15 - 10 x = 5 Answer: 5 pens can fit in the box
• Example 3: Find the roots of the quadratic equation x 2 + x - 6 = 0 Solution: Using the quadratic formula x = [-b ± √(b² - 4ac)]/2a. a = 1, b = 1, c = - 6 x = [-1 ± √(1² - 4 · 1 · -6)] / (2 · 1) x = [-1 ± √(25)] / 2 x = [-1 + 5] / 2,  [-1 - 5] / 2 x = 2, -3 Answer: The roots of the given algebraic equation are 2 and -3.

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Practice Questions on Algebraic Equations

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FAQs on Algebraic Equations

What are algebraic equations.

Algebraic equations are polynomial equations where two algebraic expressions are equated. Both sides of the equation must be balanced. The general form of an algebraic equation is P = 0.

What is an Example of Algebraic Equation?

An algebraic equation can be linear, quadratic, etc. Hence, an example of an algebraic equation can be 3x 2  - 6 = 0.

How Do You Solve Algebraic Equations?

There are many methods available to solve algebraic equations depending on the degree. Some techniques include applying simple algebraic operations , solving simultaneous equations , splitting the middle term, quadratic formula, long division, and so on.

What are Algebraic Expressions and Algebraic Equations?

Mathematical statements that consist of variables , coefficients , constants , and algebraic operations are known as algebraic expressions. When two algebraic expressions are equated together, they are known as algebraic equations.

How Do You Write Algebraic Equation?

We can convert real-life statement involving numbers and conditions into algebraic equation. For example, if the problem says, "the length of a rectangular field is 5 more than twice the width", then it can be written as the algebraic equations l = 2w + 5, where 'l' and 'w' are the length and width of the rectangular field.

What are Linear Algebraic Equations?

An algebraic equation where the highest exponent of the variable term is 1 is a linear algebraic equation. In other words, algebraic equations with degree 1 will be linear. For example, 3y - 9 = 1

Are Quadratic Equations Algebraic Equations?

Yes, quadratic equations are algebraic equations. It consists of an algebraic expression of the second degree.

What are the Basic Formulas of Algebraic Equations?

Some of the basic formulas of algebraic equations are listed below:

• Quadratic Formula: [-b ± √(b² - 4ac)]/2a
• Discriminant: b 2 - 4ac

What are the Rules for Algebraic Equations?

There are 5 basic rules for algebraic equations. These are as follows:

• Commutative Rule of Addition
• Commutative Rule of Multiplication
• Associative Rule of Addition
• Associative Rule of Multiplication
• Distributive Rule of Multiplication

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Simple Algebra Problems – Easy Exercises with Solutions for Beginners

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Understanding Algebraic Expressions

Breaking down algebra problems, solving algebraic equations, tackling algebra word problems, types of algebraic equations, algebra for different grades.

For instance, solving the equation (3x = 7) for (x) helps us understand how to isolate the variable to find its value.

I always find it fascinating how algebra serves as the foundation for more advanced topics in mathematics and science. Starting with basic problems such as ( $(x-1)^2 = [4\sqrt{(x-4)}]^2$ ) allows us to grasp key concepts and build the skills necessary for tackling more complex challenges.

So whether you’re refreshing your algebra skills or just beginning to explore this mathematical language, let’s dive into some examples and solutions to demystify the subject. Trust me, with a bit of practice, you’ll see algebra not just as a series of problems, but as a powerful tool that helps us solve everyday puzzles.

Simple Algebra Problems and Strategies

When I approach simple algebra problems, one of the first things I do is identify the variable.

The variable is like a placeholder for a number that I’m trying to find—a mystery I’m keen to solve. Typically represented by letters like ( x ) or ( y ), variables allow me to translate real-world situations into algebraic expressions and equations.

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like ( x ) or ( y )), and operators (like add, subtract, multiply, and divide). For example, ( 4x + 7 ) is an algebraic expression where ( x ) is the variable and the numbers ( 4 ) and ( 7 ) are terms. It’s important to manipulate these properly to maintain the equation’s balance.

Solving algebra problems often starts with simplifying expressions. Here’s a simple method to follow:

• Combine like terms : Terms that have the same variable can be combined. For instance, ( 3x + 4x = 7x ).
• Isolate the variable : Move the variable to one side of the equation. If the equation is ( 2x + 5 = 13 ), my job is to get ( x ) by itself by subtracting ( 5 ) from both sides, giving me ( 2x = 8 ).

With algebraic equations, the goal is to solve for the variable by performing the same operation on both sides. Here’s a table with an example:

Algebra word problems require translating sentences into equations. If a word problem says “I have six less than twice the number of apples than Bob,” and Bob has ( b ) apples, then I’d write the expression as ( 2b – 6 ).

Understanding these strategies helps me tackle basic algebra problems efficiently. Remember, practice makes perfect, and each problem is an opportunity to improve.

In algebra, we encounter a variety of equation types and each serves a unique role in problem-solving. Here, I’ll brief you about some typical forms.

Linear Equations : These are the simplest form, where the highest power of the variable is one. They take the general form ( ax + b = 0 ), where ( a ) and ( b ) are constants, and ( x ) is the variable. For example, ( 2x + 3 = 0 ) is a linear equation.

Polynomial Equations : Unlike for linear equations, polynomial equations can have variables raised to higher powers. The general form of a polynomial equation is ( $a_nx^n + a_{n-1}x^{n-1} + … + a_2x^2 + a_1x + a_0 = 0$ ). In this equation, ( n ) is the highest power, and ( $a_n$ ), ( $a_{n-1} $), …, ( $a_0$ ) represent the coefficients which can be any real number.

• Binomial Equations : They are a specific type of polynomial where there are exactly two terms. Like ($ x^2 – 4 $), which is also the difference of squares, a common format encountered in factoring.

To understand how equations can be solved by factoring, consider the quadratic equation ( $x^2$ – 5x + 6 = 0 ). I can factor this into ( (x-2)(x-3) = 0 ), which allows me to find the roots of the equation.

Here’s how some equations look when classified by degree:

Remember, identification and proper handling of these equations are essential in algebra as they form the basis for complex problem-solving.

In my experience with algebra, I’ve found that the journey begins as early as the 6th grade, where students get their first taste of this fascinating subject with the introduction of variables representing an unknown quantity.

I’ve created worksheets and activities aimed specifically at making this early transition engaging and educational.

6th Grade :

Moving forward, the complexity of algebraic problems increases:

7th and 8th Grades :

• Mastery of negative numbers: students practice operations like ( -3 – 4 ) or ( -5 $\times$ 2 ).
• Exploring the rules of basic arithmetic operations with negative numbers.
• Worksheets often contain numeric and literal expressions that help solidify their concepts.

Advanced topics like linear algebra are typically reserved for higher education. However, the solid foundation set in these early grades is crucial. I’ve developed materials to encourage students to understand and enjoy algebra’s logic and structure.

Remember, algebra is a tool that helps us quantify and solve problems, both numerical and abstract. My goal is to make learning these concepts, from numbers to numeric operations, as accessible as possible, while always maintaining a friendly approach to education.

I’ve walked through various simple algebra problems to help establish a foundational understanding of algebraic concepts. Through practice, you’ll find that these problems become more intuitive, allowing you to tackle more complex equations with confidence.

Remember, the key steps in solving any algebra problem include:

• Identifying variables and what they represent.
• Setting up the equation that reflects the problem statement.
• Applying algebraic rules such as the distributive property ($a(b + c) = ab + ac$), combining like terms, and inverse operations.
• Checking your solutions by substituting them back into the original equations to ensure they work.

As you continue to engage with algebra, consistently revisiting these steps will deepen your understanding and increase your proficiency. Don’t get discouraged by mistakes; they’re an important part of the learning process.

I hope that the straightforward problems I’ve presented have made algebra feel more manageable and a little less daunting. Happy solving!

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• Sets & Set Theory
• Trigonometry

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Math Equation Solver | Order of Operations

Calculator use.

Solve math problems using order of operations like PEMDAS, BEDMAS, BODMAS, GEMDAS and MDAS. ( PEMDAS Caution ) This calculator solves math equations that add, subtract, multiply and divide positive and negative numbers and exponential numbers. You can also include parentheses and numbers with exponents or roots in your equations.

Use these math symbols:

+  Addition -  Subtraction *  Multiplication /  Division ^  Exponents (2^5 is 2 raised to the power of 5) r  Roots (2r3 is the 3rd root of 2) () [] {}  Brackets or Grouping

You can try to copy equations from other printed sources and paste them here and, if they use ÷ for division and × for multiplication, this equation calculator will try to convert them to / and * respectively but in some cases you may need to retype copied and pasted symbols or even full equations.

If your equation has fractional exponents or roots be sure to enclose the fractions in parentheses. For example:

• 5^(2/3) is 5 raised to the 2/3
• 5r(1/4) is the 1/4 root of 5 which is the same as 5 raised to the 4th power

Entering fractions

If you want an entry such as 1/2 to be treated as a fraction then enter it as (1/2). For example, in the equation 4 divided by ½ you must enter it as 4/(1/2). Then the division 1/2 = 0.5 is performed first and 4/0.5 = 8 is performed last. If you incorrectly enter it as 4/1/2 then it is solved 4/1 = 4 first then 4/2 = 2 last. 2 is a wrong answer. 8 was the correct answer.

Math Order of Operations - PEMDAS, BEDMAS, BODMAS, GEMDAS, MDAS

PEMDAS is an acronym that may help you remember order of operations for solving math equations. PEMDAS is typcially expanded into the phrase, "Please Excuse My Dear Aunt Sally." The first letter of each word in the phrase creates the PEMDAS acronym. Solve math problems with the standard mathematical order of operations, working left to right:

• Parentheses, Brackets, Grouping - working left to right in the equation, find and solve expressions in parentheses first; if you have nested parentheses then work from the innermost to outermost
• Exponents and Roots - working left to right in the equation, calculate all exponential and root expressions second
• Multiplication and Division - next, solve both multiplication AND division expressions as they occur, working left to right in the equation. For the MDAS rule, you'll start with this step.
• Addition and Subtraction - next, solve both addition AND subtraction expressions as they occur, working left to right in the equation

PEMDAS Caution

Multiplication DOES NOT always get performed before Division. Multiplication and Division are performed as they occur in the equation, from left to right.

Addition DOES NOT always get performed before Subtraction. Addition and Subtraction are performed as they occur in the equation, from left to right.

The order "MD" (DM in BEDMAS) is sometimes confused to mean that Multiplication happens before Division (or vice versa). However, multiplication and division have the same precedence. In other words, multiplication and division are performed during the same step from left to right. For example, 4/2*2 = 4 and 4/2*2 does not equal 1.

The same confusion can also happen with "AS" however, addition and subtraction also have the same precedence and are performed during the same step from left to right. For example, 5 - 3 + 2 = 4 and 5 - 3 + 2 does not equal 0.

A way to remember this could be to write PEMDAS as PE(MD)(AS) or BEDMAS as BE(DM)(AS).

Order of Operations Acronyms

The acronyms for order of operations mean you should solve equations in this order always working left to right in your equation.

PEMDAS stands for " P arentheses, E xponents, M ultiplication and D ivision, A ddition and S ubtraction"

You may also see BEDMAS, BODMAS, and GEMDAS as order of operations acronyms. In these acronyms, "brackets" are the same as parentheses, and "order" is the same as exponents. For GEMDAS, "grouping" is like parentheses or brackets.

BEDMAS stands for " B rackets, E xponents, D ivision and M ultiplication, A ddition and S ubtraction"

BEDMAS is similar to BODMAS.

BODMAS stands for " B rackets, O rder, D ivision and M ultiplication, A ddition and S ubtraction"

GEMDAS stands for " G rouping, E xponents, D ivision and M ultiplication, A ddition and S ubtraction"

MDAS is a subset of the acronyms above. It stands for " M ultiplication, and D ivision, A ddition and S ubtraction"

Operator Associativity

Multiplication, division, addition and subtraction are left-associative. This means that when you are solving multiplication and division expressions you proceed from the left side of your equation to the right. Similarly, when you are solving addition and subtraction expressions you proceed from left to right.

Examples of left-associativity:

• a / b * c = (a / b) * c
• a + b - c = (a + b) - c

Exponents and roots or radicals are right-associative and are solved from right to left.

Examples of right-associativity:

• 2^3^4^5 = 2^(3^(4^5))
• 2r3^(4/5) = 2r(3^(4/5))

For nested parentheses or brackets, solve the innermost parentheses or bracket expressions first and work toward the outermost parentheses. For each expression within parentheses, follow the rest of the PEMDAS order: First calculate exponents and radicals, then multiplication and division, and finally addition and subtraction.

You can solve multiplication and division during the same step in the math problem: after solving for parentheses, exponents and radicals and before adding and subtracting. Proceed from left to right for multiplication and division. Solve addition and subtraction last after parentheses, exponents, roots and multiplying/dividing. Again, proceed from left to right for adding and subtracting.

Adding, Subtracting, Multiplying and Dividing Positive and Negative Numbers

This calculator follows standard rules to solve equations.

Rules for Addition Operations (+)

If signs are the same then keep the sign and add the numbers.

If signs are different then subtract the smaller number from the larger number and keep the sign of the larger number.

Rules for Subtraction Operations (-)

Keep the sign of the first number. Change all the following subtraction signs to addition signs. Change the sign of each number that follows so that positive becomes negative, and negative becomes positive then follow the rules for addition problems.

Rules for Multiplication Operations (* or ×)

Multiplying a negative by a negative or a positive by a positive produces a positive result. Multiplying a positive by a negative or a negative by a positive produces a negative result.

Rules for Division Operations (/ or ÷)

Similar to multiplication, dividing a negative by a negative or a positive by a positive produces a positive result. Dividing a positive by a negative or a negative by a positive produces a negative result.

Cite this content, page or calculator as:

Furey, Edward " Math Equation Solver | Order of Operations " at https://www.calculatorsoup.com/calculators/math/math-equation-solver.php from CalculatorSoup, https://www.calculatorsoup.com - Online Calculators

Last updated: October 19, 2023

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Math Word Problems

Welcome to the math word problems worksheets page at Math-Drills.com! On this page, you will find Math word and story problems worksheets with single- and multi-step solutions on a variety of math topics including addition, multiplication, subtraction, division and other math topics. It is usually a good idea to ensure students already have a strategy or two in place to complete the math operations involved in a particular question. For example, students may need a way to figure out what 7 × 8 is or have previously memorized the answer before you give them a word problem that involves finding the answer to 7 × 8.

There are a number of strategies used in solving math word problems; if you don't have a favorite, try the Math-Drills.com problem-solving strategy:

• Question : Understand what the question is asking. What operation or operations do you need to use to solve this question? Ask for help to understand the question if you can't do it on your own.
• Estimate : Use an estimation strategy, so you can check your answer for reasonableness in the evaluate step. Try underestimating and overestimating, so you know what range the answer is supposed to be in. Be flexible in rounding numbers if it will make your estimate easier.
• Strategize : Choose a strategy to solve the problem. Will you use mental math, manipulatives, or pencil and paper? Use a strategy that works for you. Save the calculator until the evaluate stage.
• Calculate : Use your strategy to solve the problem.
• Evaluate : Compare your answer to your estimate. If you under and overestimated, is the answer in the correct range. If you rounded up or down, does the answer make sense (e.g. is it a little less or a little more than the estimate). Also check with a calculator.

Most Popular Math Word Problems this Week

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• Addition Word Problems One-Step Addition Word Problems Using Single-Digit Numbers One-Step Addition Word Problems Using Two-Digit Numbers
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• Multiplication Word Problems One-Step Multiplication Word Problems up to 10 × 10
• Division Word Problems Division Facts Word Problems with Quotients from 5 to 12
• Multi-Step Word Problems Easy Multi-Step Word Problems

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'It was incredibly rewarding to tackle such difficult problems'

5/17/2024 A&S Communications

Andrew DiFabbio

Physics & Mathematics Brewster, N.Y.

Why did you choose Cornell?

Cornell has been host to a variety of discoveries and big names in my chosen field of physics, and it continues to be exceptional in that area. People like Kenneth Wilson, Richard Feynman, Freeman Dyson and Hans Bethe, number among the many great thinkers who have pushed the boundaries of physics at Cornell. This is not to mention Cornell’s excellence in experimental physics, having seen the discovery of such phenomena as superfluidity in 3He and hosting such facilities as the Cornell High Energy Synchrotron Source (CHESS) and the Cornell Center for Materials Research (CCMR), among others. As I see it, there is no better place for an undergraduate to study physics, including its many sub-disciplines.

What is your main extracurricular activity and why is it important to you? 

I am a member of the Big Red Marching Band. As part of the clarinet section, I have marched during football games and other events for the past three years. Having been a part of my high school’s marching band, this was one of the things that drew me to Cornell in the first place. More than anything, I appreciate the people in the band, and particularly those in my section, as a community that I can be a part of.

What Cornell memory do you treasure the most?  

During my freshman fall, when I took PHYS 1116 as my first physics course, I distinctly remember a bonus problem from the second problem set. It had to do with pursuit curves and described a scenario where one person chases another, with each moving according to the other’s position. Trying to solve the problem naively was very difficult, so the solution wasn’t immediately obvious. I remember going for a walk through the arts quad and looking up at the clock tower when it occurred to me that the problem admitted a simple geometric solution, and I went back to my dorm room on north campus to write it up. I could have chosen a number of different occasions like this, when a problem that seems impossible suddenly becomes trivial, but this was one of the first times I had that experience at Cornell, so I remember it fondly.       

What have you accomplished as a Cornell student that you are most proud of?

I was involved in research under two different faculty members, Professor Rob Thorne and Professor Carl Franck. My work with Rob Thorne concerned cryogenic electron microscopy, particularly the method of single particle analysis. Cryo-EM seeks to resolve the structure of protein complexes, typically to around 3Å resolution, and allows observing them in conformational states more faithful to their natural environments than is achievable with techniques like protein crystallography. Under Carl Franck, my work involved developing code to process data from a coincidence experiment; I vastly improved the performance of our data reduction algorithm and, along the way, developed a numerical method to characterize the time resolution of the detectors involved. I developed a variety of skills from these experiences, including cryo-EM sample preparation and familiarity with several numerical techniques. It was incredibly rewarding to be involved in the scientific process and to tackle such difficult problems.

How have your beliefs or perspectives changed since you first arrived at Cornell? 

On an intellectual level, I’ve always understood that sharing people’s company and perspectives is important for academia, just as it is in everyday life. However, my behavior didn’t always reflect this fact. In high school, it was fairly easy to be an island and work on everything alone; I’ve since realized the limits of this approach and understand the value in collaboration that simply can’t be achieved individually.

Every year, our faculty nominate graduating Arts & Sciences students to be featured as part of our Extraordinary Journeys series.  Read more about the Class of 202 4.

Diversity, empathy fuel national win for Men’s Fencing Club

'Applicable Algebra reshaped my understanding of math'

'I studied why miscommunication and ambiguity are so prevalent in human language'

'I've met a variety of people with a wide range of interests'

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Praxis Core Math

Course: praxis core math   >   unit 1.

• Algebraic properties | Lesson
• Algebraic properties | Worked example
• Solution procedures | Lesson
• Solution procedures | Worked example
• Equivalent expressions | Lesson
• Equivalent expressions | Worked example
• Creating expressions and equations | Lesson
• Creating expressions and equations | Worked example

Algebraic word problems | Lesson

• Algebraic word problems | Worked example
• Linear equations | Lesson
• Linear equations | Worked example
• Quadratic equations | Lesson
• Quadratic equations | Worked example

What are algebraic word problems?

What skills are needed.

• Translating sentences to equations
• Solving linear equations with one variable
• Evaluating algebraic expressions
• Solving problems using Venn diagrams

How do we solve algebraic word problems?

• Define a variable.
• Write an equation using the variable.
• Solve the equation.
• If the variable is not the answer to the word problem, use the variable to calculate the answer.

What's a Venn diagram?

• 7 + 10 − 13 = 4 ‍   brought both food and drinks.
• 7 − 4 = 3 ‍   brought only food.
• 10 − 4 = 6 ‍   brought only drinks.
• Your answer should be
• an integer, like 6 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  
• an exact decimal, like 0.75 ‍  
• a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍  
• (Choice A)   $ 4 ‍   A $ 4 ‍  
• (Choice B)   $ 5 ‍   B $ 5 ‍  
• (Choice C)   $ 9 ‍   C $ 9 ‍  
• (Choice D)   $ 14 ‍   D $ 14 ‍  
• (Choice E)   $ 20 ‍   E $ 20 ‍  
• (Choice A)   10 ‍   A 10 ‍  
• (Choice B)   12 ‍   B 12 ‍  
• (Choice C)   24 ‍   C 24 ‍  
• (Choice D)   30 ‍   D 30 ‍  
• (Choice E)   32 ‍   E 32 ‍  
• (Choice A)   4 ‍   A 4 ‍  
• (Choice B)   10 ‍   B 10 ‍  
• (Choice C)   14 ‍   C 14 ‍  
• (Choice D)   18 ‍   D 18 ‍  
• (Choice E)   22 ‍   E 22 ‍  

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• Pre Algebra Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios & Proportions Percent Modulo Number Line Expanded Form Mean, Median & Mode
• Algebra Equations Inequalities System of Equations System of Inequalities Basic Operations Algebraic Properties Partial Fractions Polynomials Rational Expressions Sequences Power Sums Interval Notation Pi (Product) Notation Induction Logical Sets Word Problems
• Pre Calculus Equations Inequalities Scientific Calculator Scientific Notation Arithmetics Complex Numbers Polar/Cartesian Simultaneous Equations System of Inequalities Polynomials Rationales Functions Arithmetic & Comp. Coordinate Geometry Plane Geometry Solid Geometry Conic Sections Trigonometry
• Calculus Derivatives Derivative Applications Limits Integrals Integral Applications Integral Approximation Series ODE Multivariable Calculus Laplace Transform Taylor/Maclaurin Series Fourier Series Fourier Transform
• Functions Line Equations Functions Arithmetic & Comp. Conic Sections Transformation
• Linear Algebra Matrices Vectors
• Trigonometry Identities Proving Identities Trig Equations Trig Inequalities Evaluate Functions Simplify
• Statistics Mean Geometric Mean Quadratic Mean Average Median Mode Order Minimum Maximum Probability Mid-Range Range Standard Deviation Variance Lower Quartile Upper Quartile Interquartile Range Midhinge Standard Normal Distribution
• Physics Mechanics
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• Finance Simple Interest Compound Interest Present Value Future Value
• Economics Point of Diminishing Return
• Conversions Roman Numerals Radical to Exponent Exponent to Radical To Fraction To Decimal To Mixed Number To Improper Fraction Radians to Degrees Degrees to Radians Hexadecimal Scientific Notation Distance Weight Time Volume
• Pre Algebra
• One-Step Addition
• One-Step Subtraction
• One-Step Multiplication
• One-Step Division
• One-Step Decimals
• Two-Step Integers
• Two-Step Add/Subtract
• Two-Step Multiply/Divide
• Two-Step Fractions
• Two-Step Decimals
• Multi-Step Integers
• Multi-Step with Parentheses
• Multi-Step Rational
• Multi-Step Fractions
• Multi-Step Decimals
• Solve by Factoring
• Completing the Square
• Quadratic Formula
• Biquadratic
• Logarithmic
• Exponential
• Rational Roots
• Floor/Ceiling
• Equation Given Roots
• Newton Raphson
• Substitution
• Elimination
• Cramer's Rule
• Gaussian Elimination
• System of Inequalities
• Perfect Squares
• Difference of Squares
• Difference of Cubes
• Sum of Cubes
• Polynomials
• Distributive Property
• FOIL method
• Perfect Cubes
• Binomial Expansion
• Negative Rule
• Product Rule
• Quotient Rule
• Expand Power Rule
• Fraction Exponent
• Exponent Rules
• Exponential Form
• Logarithmic Form
• Absolute Value
• Rational Number
• Powers of i
• Complex Form
• Partial Fractions
• Is Polynomial
• Leading Coefficient
• Leading Term
• Standard Form
• Complete the Square
• Synthetic Division
• Linear Factors
• Rationalize Denominator
• Rationalize Numerator
• Identify Type
• Convergence
• Interval Notation
• Pi (Product) Notation
• Boolean Algebra
• Truth Table
• Mutual Exclusive
• Cardinality
• Caretesian Product
• Age Problems
• Distance Problems
• Cost Problems
• Investment Problems
• Number Problems
• Percent Problems
• Addition/Subtraction
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• Pre Calculus
• Linear Algebra
• Trigonometry
• Conversions

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Number line.

• x^4-5x^2+4=0
• \sqrt{x-1}-x=-7
• \left|3x+1\right|=4
• \log _2(x+1)=\log _3(27)
• 3^x=9^{x+5}
• What is the completing square method?
• Completing the square method is a technique for find the solutions of a quadratic equation of the form ax^2 + bx + c = 0. This method involves completing the square of the quadratic expression to the form (x + d)^2 = e, where d and e are constants.
• What is the golden rule for solving equations?
• The golden rule for solving equations is to keep both sides of the equation balanced so that they are always equal.
• How do you simplify equations?
• To simplify equations, combine like terms, remove parethesis, use the order of operations.
• How do you solve linear equations?
• To solve a linear equation, get the variable on one side of the equation by using inverse operations.

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• High School Math Solutions – Exponential Equation Calculator Solving exponential equations is pretty straightforward; there are basically two techniques: <ul> If the exponents...

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Watch CBS News

Teens come up with trigonometry proof for Pythagorean Theorem, a problem that stumped math world for centuries

By Bill Whitaker

May 5, 2024 / 7:00 PM EDT / CBS News

As the school year ends, many students will be only too happy to see math classes in their rearview mirrors. It may seem to some of us non-mathematicians that geometry and trigonometry were created by the Greeks as a form of torture, so imagine our amazement when we heard two high school seniors had proved a mathematical puzzle that was thought to be impossible for 2,000 years. 

We met Calcea Johnson and Ne'Kiya Jackson at their all-girls Catholic high school in New Orleans. We expected to find two mathematical prodigies.

Instead, we found at St. Mary's Academy , all students are told their possibilities are boundless.

Come Mardi Gras season, New Orleans is alive with colorful parades, replete with floats, and beads, and high school marching bands.

In a city where uniqueness is celebrated, St. Mary's stands out – with young African American women playing trombones and tubas, twirling batons and dancing - doing it all, which defines St. Mary's, students told us.

Junior Christina Blazio says the school instills in them they have the ability to accomplish anything. 

Christina Blazio: That is kinda a standard here. So we aim very high - like, our aim is excellence for all students. 

The private Catholic elementary and high school sits behind the Sisters of the Holy Family Convent in New Orleans East. The academy was started by an African American nun for young Black women just after the Civil War. The church still supports the school with the help of alumni.

In December 2022, seniors Ne'Kiya Jackson and Calcea Johnson were working on a school-wide math contest that came with a cash prize.

Ne'Kiya Jackson: I was motivated because there was a monetary incentive.

Calcea Johnson: 'Cause I was like, "$500 is a lot of money. So I-- I would like to at least try."

Both were staring down the thorny bonus question.

Bill Whitaker: So tell me, what was this bonus question?

Calcea Johnson: It was to create a new proof of the Pythagorean Theorem. And it kind of gave you a few guidelines on how would you start a proof.

The seniors were familiar with the Pythagorean Theorem, a fundamental principle of geometry. You may remember it from high school: a² + b² = c². In plain English, when you know the length of two sides of a right triangle, you can figure out the length of the third.

Both had studied geometry and some trigonometry, and both told us math was not easy. What no one told  them  was there had been more than 300 documented proofs of the Pythagorean Theorem using algebra and geometry, but for 2,000 years a proof using trigonometry was thought to be impossible, … and that was the bonus question facing them.

Bill Whitaker: When you looked at the question did you think, "Boy, this is hard"?

Ne'Kiya Jackson: Yeah. 

Bill Whitaker: What motivated you to say, "Well, I'm going to try this"?

Calcea Johnson: I think I was like, "I started something. I need to finish it." 

Bill Whitaker: So you just kept on going.

Calcea Johnson: Yeah.

For two months that winter, they spent almost all their free time working on the proof.

CeCe Johnson: She was like, "Mom, this is a little bit too much."

CeCe and Cal Johnson are Calcea's parents.

CeCe Johnson:   So then I started looking at what she really was doing. And it was pages and pages and pages of, like, over 20 or 30 pages for this one problem.

Cal Johnson: Yeah, the garbage can was full of papers, which she would, you know, work out the problems and-- if that didn't work she would ball it up, throw it in the trash. 

Bill Whitaker: Did you look at the problem? 

Neliska Jackson is Ne'Kiya's mother.

Neliska Jackson: Personally I did not. 'Cause most of the time I don't understand what she's doing (laughter).

Michelle Blouin Williams: What if we did this, what if I write this? Does this help? ax² plus ….

Their math teacher, Michelle Blouin Williams, initiated the math contest.

Bill Whitaker: And did you think anyone would solve it?

Michelle Blouin Williams: Well, I wasn't necessarily looking for a solve. So, no, I didn't—

Bill Whitaker: What were you looking for?

Michelle Blouin Williams: I was just looking for some ingenuity, you know—

Calcea and Ne'Kiya delivered on that! They tried to explain their groundbreaking work to 60 Minutes. Calcea's proof is appropriately titled the Waffle Cone.

Calcea Johnson: So to start the proof, we start with just a regular right triangle where the angle in the corner is 90°. And the two angles are alpha and beta.

Bill Whitaker: Uh-huh

Calcea Johnson: So then what we do next is we draw a second congruent, which means they're equal in size. But then we start creating similar but smaller right triangles going in a pattern like this. And then it continues for infinity. And eventually it creates this larger waffle cone shape.

Calcea Johnson: Am I going a little too—

Bill Whitaker: You've been beyond me since the beginning. (laughter) 

Bill Whitaker: So how did you figure out the proof?

Ne'Kiya Jackson: Okay. So you have a right triangle, 90° angle, alpha and beta.

Bill Whitaker: Then what did you do?

Ne'Kiya Jackson: Okay, I have a right triangle inside of the circle. And I have a perpendicular bisector at OP to divide the triangle to make that small right triangle. And that's basically what I used for the proof. That's the proof.

Bill Whitaker: That's what I call amazing.

Ne'Kiya Jackson: Well, thank you.

There had been one other documented proof of the theorem using trigonometry by mathematician Jason Zimba in 2009 – one in 2,000 years. Now it seems Ne'Kiya and Calcea have joined perhaps the most exclusive club in mathematics. 

Bill Whitaker: So you both independently came up with proof that only used trigonometry.

Ne'Kiya Jackson: Yes.

Bill Whitaker: So are you math geniuses?

Calcea Johnson: I think that's a stretch. 

Bill Whitaker: If not genius, you're really smart at math.

Ne'Kiya Jackson: Not at all. (laugh) 

To document Calcea and Ne'Kiya's work, math teachers at St. Mary's submitted their proofs to an American Mathematical Society conference in Atlanta in March 2023.

Ne'Kiya Jackson: Well, our teacher approached us and was like, "Hey, you might be able to actually present this," I was like, "Are you joking?" But she wasn't. So we went. I got up there. We presented and it went well, and it blew up.

Bill Whitaker: It blew up.

Calcea Johnson: Yeah. 

Ne'Kiya Jackson: It blew up.

Bill Whitaker: Yeah. What was the blowup like?

Calcea Johnson: Insane, unexpected, crazy, honestly.

It took millenia to prove, but just a minute for word of their accomplishment to go around the world. They got a write-up in South Korea and a shout-out from former first lady Michelle Obama, a commendation from the governor and keys to the city of New Orleans. 

Bill Whitaker: Why do you think so many people found what you did to be so impressive?

Ne'Kiya Jackson: Probably because we're African American, one. And we're also women. So I think-- oh, and our age. Of course our ages probably played a big part.

Bill Whitaker: So you think people were surprised that young African American women, could do such a thing?

Calcea Johnson: Yeah, definitely.

Ne'Kiya Jackson: I'd like to actually be celebrated for what it is. Like, it's a great mathematical achievement.

Achievement, that's a word you hear often around St. Mary's academy. Calcea and Ne'Kiya follow a long line of barrier-breaking graduates. 

The late queen of Creole cooking, Leah Chase , was an alum. so was the first African-American female New Orleans police chief, Michelle Woodfork …

And judge for the Fifth Circuit Court of Appeals, Dana Douglas. Math teacher Michelle Blouin Williams told us Calcea and Ne'Kiya are typical St. Mary's students.  

Bill Whitaker: They're not unicorns.

Michelle Blouin Williams: Oh, no no. If they are unicorns, then every single lady that has matriculated through this school is a beautiful, Black unicorn.

Pamela Rogers: You're good?

Pamela Rogers, St. Mary's president and interim principal, told us the students hear that message from the moment they walk in the door.

Pamela Rogers: We believe all students can succeed, all students can learn. It does not matter the environment that you live in. 

Bill Whitaker: So when word went out that two of your students had solved this almost impossible math problem, were they universally applauded?

Pamela Rogers: In this community, they were greatly applauded. Across the country, there were many naysayers.

Bill Whitaker: What were they saying?

Pamela Rogers: They were saying, "Oh, they could not have done it. African Americans don't have the brains to do it." Of course, we sheltered our girls from that. But we absolutely did not expect it to come in the volume that it came.  

Bill Whitaker: And after such a wonderful achievement.

Pamela Rogers: People-- have a vision of who can be successful. And-- to some people, it is not always an African American female. And to us, it's always an African American female.

Gloria Ladson-Billings: What we know is when teachers lay out some expectations that say, "You can do this," kids will work as hard as they can to do it.

Gloria Ladson-Billings, professor emeritus at the University of Wisconsin, has studied how best to teach African American students. She told us an encouraging teacher can change a life.

Bill Whitaker: And what's the difference, say, between having a teacher like that and a whole school dedicated to the excellence of these students?

Gloria Ladson-Billings: So a whole school is almost like being in Heaven. 

Bill Whitaker: What do you mean by that?

Gloria Ladson-Billings: Many of our young people have their ceilings lowered, that somewhere around fourth or fifth grade, their thoughts are, "I'm not going to be anything special." What I think is probably happening at St. Mary's is young women come in as, perhaps, ninth graders and are told, "Here's what we expect to happen. And here's how we're going to help you get there."

At St. Mary's, half the students get scholarships, subsidized by fundraising to defray the $8,000 a year tuition. Here, there's no test to get in, but expectations are high and rules are strict: no cellphones, modest skirts, hair must be its natural color.

Students Rayah Siddiq, Summer Forde, Carissa Washington, Tatum Williams and Christina Blazio told us they appreciate the rules and rigor.

Rayah Siddiq: Especially the standards that they set for us. They're very high. And I don't think that's ever going to change.

Bill Whitaker: So is there a heart, a philosophy, an essence to St. Mary's?

Summer Forde: The sisterhood—

Carissa Washington: Sisterhood.

Tatum Williams: Sisterhood.

Bill Whitaker: The sisterhood?

Voices: Yes.

Bill Whitaker: And you don't mean the nuns. You mean-- (laughter)

Christina Blazio: I mean, yeah. The community—

Bill Whitaker: So when you're here, there's just no question that you're going to go on to college.

Rayah Siddiq: College is all they talk about. (laughter) 

Pamela Rogers: … and Arizona State University (Cheering)

Principal Rogers announces to her 615 students the colleges where every senior has been accepted.

Bill Whitaker: So for 17 years, you've had a 100% graduation rate—

Pamela Rogers: Yes.

Bill Whitaker: --and a 100% college acceptance rate?

Pamela Rogers: That's correct.

Last year when Ne'Kiya and Calcea graduated, all their classmates went to college and got scholarships. Ne'Kiya got a full ride to the pharmacy school at Xavier University in New Orleans. Calcea, the class valedictorian, is studying environmental engineering at Louisiana State University.

Bill Whitaker: So wait a minute. Neither one of you is going to pursue a career in math?

Both: No. (laugh)

Calcea Johnson: I may take up a minor in math. But I don't want that to be my job job.

Ne'Kiya Jackson: Yeah. People might expect too much out of me if (laugh) I become a mathematician. (laugh)

But math is not completely in their rear-view mirrors. This spring they submitted their high school proofs for final peer review and publication … and are still working on further proofs of the Pythagorean Theorem. Since their first two …

Calcea Johnson: We found five. And then we found a general format that could potentially produce at least five additional proofs.

Bill Whitaker: And you're not math geniuses?

Bill Whitaker: I'm not buying it. (laughs)

Produced by Sara Kuzmarov. Associate producer, Mariah B. Campbell. Edited by Daniel J. Glucksman.

Bill Whitaker is an award-winning journalist and 60 Minutes correspondent who has covered major news stories, domestically and across the globe, for more than four decades with CBS News.

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• a special character: @$#!%*?&

ChatGPT may be about to make your job way easier

• You'll soon be able to pull ChatGPT up on your screen and get real-time feedback on your work.
• ChatGPT was able to suggest fixes to errors in code in a live demo on Monday.
• OpenAI debuted the feature as part of GPT-4o, the latest model powering the AI tool.

The latest version of ChatGPT could turn the artificial-intelligence tool into a personal assistant — including at work .

Among the new features that OpenAI highlighted on Monday: You can pull up ChatGPT on your computer desktop or smartphone screen and show it whatever you're looking at, whether it's a math problem on a whiteboard or a screen full of code.

In one demo during OpenAI's presentation on its newest model powering the chatbot, GPT-4o (that's the letter O, not zero), ChatGPT suggested changes to code in real time on a user's computer.

At first, ChatGPT was known for its command-prompt-like interface, which allows users to type in questions or requests for images. Last fall, OpenAI added a voice mode, allowing users to ask questions like they would with Alexa or Siri — albeit with a more sophisticated model responding.

But with GPT-4o, users can now show ChatGPT what they're working on and get real-time feedback.

Related stories

In another example, ChatGPT gave a one-sentence summary of a line graph when an OpenAI employee pulled it up on his desktop.

Damn... the idea of being able to just share your screen natively using the new ChatGPT Mac app and have gpt work with you is freaking awesome 🤯 pic.twitter.com/mwZvZWVP1u — Josh Cohenzadeh (@jshchnz) May 13, 2024

With the GPT-4o/ChatGPT desktop app, you can have a coding buddy (black circle) that talks to you and sees what you see! #openai announcements thread! https://t.co/CpvCkjI0iA pic.twitter.com/Tfh81mBHCv — Andrew Gao (@itsandrewgao) May 13, 2024

For anyone who codes or handles data for a living, it's not hard to imagine how the latest version of ChatGPT could make work easier .

Holy shit ChatGPT can help you fix and write codes in real freaking time, it can see your screen and can talk to you in real time. I hope you got an idea about how crazy this shit is pic.twitter.com/ZxqsL2TOje — Kuldeep Lather (@Kullthegreat) May 13, 2024

ChatGPT can also give feedback on work in the real world. In Monday's demo, OpenAI employees showed the tool, via smartphone camera, a simple algebra problem written on a whiteboard.

Eventually, ChatGPT gave them hints on how to solve an unknown variable, giving them feedback at different stages.

GPT-4o marks "the first time that we are really making a huge step forward when it comes to the ease of use," OpenAI's chief technology officer, Mira Murati, told an audience at the company's offices as Monday's presentation kicked off.

Correction: May 13, 2024 — A previous version of this story referred to the latest version of ChatGPT by the wrong name. It is still called ChatGPT, not ChatGPT-4o. GPT-4o is the new model that powers the chatbot.

Axel Springer, Business Insider's parent company, has a global deal to allow OpenAI to train its models on its media brands' reporting.

Watch: What is ChatGPT, and should we be afraid of AI chatbots?

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