# 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.

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

#### 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.

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

##### Practice 1

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

##### Practice 2

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

##### Practice 3

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

##### Practice 4

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

##### Practice 5

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.
##### Practice 6

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

##### Practice 7

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