Solving Quadratic Equations

A quadratic equation in x in standard form is an equation that can be written in the form ax + bc + c = 0. It is called a second-degree equation because the degree of its variable is 2. If a quadratic equation has solutions, there will be at most two solutions.There could be none or only one real solution.

Soving by factoring

Many times you can use factoring and the zero factor property to solve a quadratic equation. This property states that if uv=0 then either u = 0 or v = 0! That means, if you multiply two numbers together and get zero, one of the numbers has to be zero!!

Your first attempt to solve any quadratic equation should be to try to factor it!

EXAMPLE 1:

Solve the following quadratic equation

x -x - 12 = 0

(x-4) (x+3) = 0 First try to factor it

Either x - 4 = 0 or x + 3 = 0 Now use the zero property

x = 4 , x = -3

Check both solutions:

Check for x = 4	Check for x = -3
x -x - 12 = 0	x -x - 12 = 0
(4)2 -(4) -12 = 0	(-3)2 -(-3) -12 = 0
16 -4 -12 = 0	9 + 3 -12 = 0
0 = 0	0 = 0

EXAMPLE 2:

Solve the following quadratic equation

10x + x - 3 = 0

(5x + 3)(2x - 1) = 0 First try to factor it !!!

Either 5x + 3 = 0 or 2x - 1 = 0 Now use the zero property

5x = -3 or 2x = 1

x = -3/5 , x = 1/2

A quadratic equation can have repeated solutions

EXAMPLE 3:

Solve x + 10x + 25 = 0

(x+5)(x+5) = 0

So x + 5 = 0 or x + 5 = 0

x = -5 or x = -5

This solution is called a repeated solution since both factors were the same! In this case, you would only have to check one solution.

EXAMPLE 4:

Be careful on the type of problems! To solve a quadratic equation using the zero property, the right side of the equation must be zero!

Solve for x:

(x-3)(x+7)=24

x +7x - 3x -21 = 24 First simplify

x +7x -3x -21 -24 = 0 Move the terms around - leave nothing on the right side

x +4x -45 = 0 Collect terms

(x+9)(x-5) = 0 Factor

So x+9 = 0 or x-5 = 0 Use the zero property

x = -9 or x = 5

Checks: