Solve a quadratic equation by factoring

Step by Step Examples

What is a Quadratic Equation?

A quadratic equation is an equation that can be written as:

$$ ax^2 + bx + c $$ where a ≠ 0.

A quadratic equation must have a squared term as its highest power.

Examples of quadratic equations

$$ y = 5x^2 + 2x + 5 \\ y = 11x^2 + 22 \\ y = x^2 - 4x +5 \\ y = -x^2 + 5 $$

Non Examples

$$ y = 11x + 22 \\ y = x^3 -x^2 +5x +5 \\ y = 2x^3 -4x^2 \\ y = -x^4 + 5 $$

What is a the solution of a Quadratic Equation?

The solution of a quadratic equation is the value of x when you set the equation equal to $$ \red {\text {zero}}$$
i.e. When you solve the following general equation: $$\red 0 = ax^2 + bx + c $$.

There are many ways to solve quadratic equations. One of the ways is to factor the equation.

General Steps to solve by factoring

Create a factor chart for all factor pairs of c.

A factor pair is just two numbers that multiply and give you c.

  1. Out of all of the factor pairs from step 1, look for the pair (if it exists) that add up to b

    Note: if the pair does not exist, you must either complete the square or use the quadratic formula.

  2. Insert the pair you found in step 2 into two binomals.

Solve each binomial for zero to get the solutions of the quadratic equation.

Example of how to solve a quadratic equation by factoring

Quadratic Equation: y = x² + 2x + 1.

Picture of steps to solve quadratic equation by factoring

Below is a picture representing the graph of y = x² + 2x + 1 as well as the solution we found by factoring.

Picture of graph of solved quadratic formula
Practice 1

Use the steps above to solve the quadratic equation by factoring.

Quadratic equation: $$ y = x^2 - 2x + 1 $$.

picture steps solve factor

Below is a picture of the graph of the quadratic $$ y = x^2 - 2x + 1 $$as well as the solutions.

solution 1 by 0
Practice 2

Calculate the solutions of the quadratic equation below by factoring quadratic equation: $$ y = x^2 + 4x + 4 $$.

Steps to solve quadratic equation by factoring

Below is a picture of the graph of the quadratic $$ y = x^2 + 4x + 4 $$ as well as the solutions.

solution -2 and 0
Practice 3

Calculate the solutions of the quadratic equation below by factoring.

Quadratic equation: $$ y = x^2 - 4x + 4 $$ .

Steps to solve quadratic by factoring

$$ y = x^2 - 4x + 4 $$ is graphed below as well as its solution (2, 0).

solution 2 and 0
Practice 4

Calculate the solutions of the quadratic equation below by factoring.

Quadratic equation: $$ y = x^2 + 6x + 9$$ .

Steps to solve quadratic by factoring

$$ y = x^2 + 6x + 9 $$ is graphed below as well as its solution (2, 0).

solution -3 and 0
Practice 5

Calculate the solutions of the quadratic equation below by factoring.

Quadratic: $$ y = x^2 - 6x + 9 $$.

y = x² − 6x + 9

  • c = 9
  • b = 6
  • The only factors of c whose sum is b are -3 • -3.
  • y = (x − 3)(x − 3)
  • 0 = (x − 3)
  • The solution is at x = 3.
solution 3 and 0
Practice 6

Use the steps to solve the equation below.

Quadratic equation: $$ y = x^2 + 2x - 3$$ .

Steps to solve quadratic by factoring

$$ y = x^2 + 2x - 3$$ is graphed below as well as both the solutions.

solution -3 and 0 and 1 and 0
Practice 7

Use the steps to solve the equation below.

Quadratic equation: $$ y = x^2 - 2x - 3 $$ .

Steps to solve quadratic by factoring

$$ y = x^2 - 2x - 3 $$ is graphed below as well as both the solutions.

solution 3 and 0 and -1 and 0
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