Quadratic equations can seem intimidating, but they don’t have to be!

Learning to solve equations by factoring is a powerful tool that can make math much easier and even a bit fun.

Learn to break it down step-by-step, so you can confidently tackle any quadratic equation that comes your way!

Aboard with the quadratic equation ax squared + bx + c = 0

Essentials of Quadratic Equations

A quadratic equation is a polynomial equation of degree two, typically written in the form ax² + bx + c = 0, where a and b are coefficients, c is a constant, and a isn't zero.

A parabola graph showing x-intercepts and y-intercepts. Image courtesy of mathspace

Standard Form:

Standard form ax^2 + bx + c where a is the coefficient of x^2, b is the coefficient of x, and c is the constant and y-int.

Factored Form:

Factored form a(x-r)(x-s)=0 with a as the coefficient and r and s as the roots (x-intercepts).

Images created by the author using Google Slides

Solving Quadratics:

To solve quadratic equations, the goal is to get the equation into factored form and find what x is equal to. To get to factored form from standard form, you'll use a method called factoring.

Factoring is the process if breaking down an expression or number into factors.

Preparing to Factor

There are a few things to consider before you solve equations by factoring:

  1. Always be to factor out the Greatest Common Factor, if there is one.

  2. If a = 1, you can use the Sum-Product Method.

  3. If a ≠ 1, you can use Decomposition.

  4. There are special cases of Perfect Square Trinomials and Difference of Squares that have their own rules.

  5. The Quadratic Formula will always give you the value of x if you're unable to factor.

A word cloud of factoring, gcf, sum-product, decomposition, grouping, perfect squares, difference of squares. Image created by the author using WordArt

1. Common Factoring

ALWAYS start any factoring method by finding the Greatest Common Factor (GCF) of the terms in the quadratic and pulling it out.

The greatest common factor is the largest factor that multiple terms share.

Step 1: Find the Greatest Common Factor. Check out How to Find the Greatest Common Factor if you need a refresher.

Step 2: Divide the terms by the GCF.

Step 3: Pull the GCF out of the expression and leave the divided terms in the brackets.

Example:

10x³ + 30x² + 55x

The GCF of 10x^3 + 30x^2 + 55x is 5x. The factored form is 5x(2x^2 + 6x + 11). Images in this Byte were created by the author using Google Slides. To hear audio descriptions of these images, click play on the audio player below each image:

Quiz

Factor the GCF out from 8x² - 12x + 16:

2. Sum-Product Method when a = 1

The Sum-Product Method is all about finding two numbers (r and s) that add up b (the coefficient of x) and multiply to c (the constant term). This method can only be used when a = 1 in ax² + bx + c.

Step 1: Find all the pairs of numbers that multiply to c. This includes negative numbers.

Step 2: Determine which pair adds up to get b.

Step 3: Plug the pair (r and s) into (x + r)(x + s) = 0.

Step 4: Find the x values that make the expression true (i.e., where x + r = 0 and x + s = 0).

Example:

x² - 7x + 12

  • a = 1

  • b = -7

  • c = 12

The factored form of x^2 - 7x + 12 is (x-3)(x-4) with the solution x = 3 and x = 4.

Quiz

Solve x² + 2x - 8 using the Sum-Product method:

3. Decomposition Method when a ≠ 1

The process of decomposition is breaking up the middle term in the expression and using grouping to fully factor. Use this method when you have more complex quadratics, such as when a ≠ 1.

Grouping: Arranging terms into smaller groups that have a common factor. If you need a reminder on how to group expressions, check out Factor a Quadratic Equation by Grouping Terms.

Step 1: Find all the pairs of numbers that multiply to (a)(c).

Step 2: Determine which factor pair (r and s) adds to get b.

Step 3: Break the middle term of the expression (bx) into rx and sx.

Step 4: Group each half of the expression and factor out the GCFS.

Step 5: Factor the GCF bracket out of the expression.

Step 6: Set the expression equal to 0 and solve for x.

Example:

Let's look at 2x² + 7x + 5

  • a = 2

  • b = 7

  • c = 5

The factored form of 2x^2 + 7x + 5 is (x+1)(2x+5) with the solution x = -2.5 and x = -1.

Quiz

Solve 7x² + 3x - 10 using decomposition.

4. Special Cases

Sometimes, quadratic equations fall into special categories that allow for quick and easy factoring. Recognizing these cases can save you time and effort!

Perfect Square Trinomials

Perfect Square Trinomials have two square terms ( and ) with the third term 2ab.

They follow the pattern:

  • a² + 2ab + b² = (a + b)²

  • a² - 2ab + b² = (a - b)²

Examples:

x^2 + 10x^2 + 25 = (x+5)^2 and 4x^2 - 28xy + 49y^2 = (2x-7y)^2.

Difference of Squares

Difference of Squares have two square terms ( and ) separated by a minus sign.

They follow the pattern:

  • a² - b² = (a + b) (a - b)

Examples:

x^2 - 9 = (x+3)(x-3) and 25x^2 - 49 = (5x+7)(5x-7).

Quiz

Solve 4x² - 36.

5. Quadratic Formula

When solving equations by factoring gets tricky or when you can't find nice numbers to work with, the quadratic formula comes to the rescue! It allows you to find the solutions for any quadratic equation, no matter how complicated.

Step 1: Sub the values of a, b, and c into the quadratic formula.

the quadratic formula: b plus minus the square root of b squared minus 4ac, all divided by 2a. Image courtesy of CUEMATH

Step 2: Solve for x.

Example:

5x² + 10x + 4

  • a = 5

  • b = 10

  • c = 4

Quadratic equation being solved (audio description below). The solution is x = -1.45, -0.55.

Quiz

Solve 6x² + 25x + 14 using the quadratic formula:

Take Action

It can be tricky to learn a new concept but the most important thing is to take it step by step and PRACTICE. Practice makes perfect!

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Not that you know how to solve quadratic equations by factoring:

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