Unit 8 - Objective 1 - Quadratic Equations - Solution by Factoring

A polynomial equation of the second degree is called a quadratic equation.

If a, b, and c are constants and a 0, then

ax2 + bx + c = 0

 1 x2 = 5 x2 - 5 = 0 a = 1, b = 0, c = -5 2 3x2 - x + 5 = 0 a= 3, b = -1, c = 5 3 -x2 = x -x2 - x = 0 a = - 1, b = - 1, c = 0

The following are not quadratic equations:

 1 4x = 5 (There is not a x2 term) 2 x3 + x2 - x + 2 = 0 (There should not be any terms with a degree higher than 2)

As you solve quadratic equations we need to know the zero principle: if a and b are numbers and ab = 0 then a=0, b=0 or both a and b are 0. To find roots, make sure the equation equals 0 to use the zero principle.

Examples:

1. x2 - x - 6 = 0
We need to factor
(x - 3) (x + 2) = 0
 So x - 3 = 0 x = 3 or x + 2 = 0 x = - 2

2. 6x2 - 11x - 35 = 0
We need to factor
(3x + 5) (2x - 7) = 0
 So 3x + 5 = 0 3x = - 5 x = -5/3 or 2x - 7 = 0 2x = 7 x = 7/2

3. x2 = 25
Set the equation equal to zero
x2 - 25 = 0
(x + 5) (x - 5) = 0
 So x + 5 = 0 x = - 5 or x - 5 = 0 x = 5

4.

You want to clear the denominator first
by multiplying both sides by the LCD
which is x (3 - 2x)

6 - 4x = x (3 - 2x)
6 - 4x = 3x - 2x2

Get everything on one side of the equation to get 0

2x2 - 7x + 6 = 0

Now to factor

(2x - 3) (x - 2) = 0

 2x - 3 = 0 2x = 3 x = 3/2 or x - 2 = 0 x = 2 x = 2