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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 = 2x² + 3x + 7
- y = 13x² + 2
- y = x² − 4x + 15
- y = −x² + 5

Examples of equations that are **not** quadratic

- y = 11x + 223
- y = x
^{3}− 4x² +5x +6 - y = 2x
^{3}− 7x² - y = −x
^{4}+ 5

The solution of a quadratic equation is the value of x when you set the equation equal to zero

i.e. When you solve the following general equation:

0 = ax² + bx + c

Given a quadratic equation:

ax ² + bx + c

One method to solve the equation for zero is to factor the equations.

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'

- 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.

- Insert the pair you found in step 2 into two binomials
- 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 + 1as well as the solution we found by factoring

Below is a picture of the graph of the quadratic

y = x² − 2x + 1as well as the solutions

Below is a picture of the graph of the quadratic y = x² − 2x + 1 as well as the solutions

y = x² − 4x − 4is graphed below as well as its solution (2,0)

xy = x² + 6x + 9is graphed below as well as its solution (2,0)

- 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

y = x² + 2x − 3is graphed below as well as both the solutions

y = x² − 2x − 3is graphed below as well as both the solutions