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 ntequ2.gif (849 bytes) 0, then

ax2 + bx + c = 0

is the general quadratic equation.  The following are quadratic equations:

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. u8obj1-01.gif (1441 bytes)

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

u8obj1-02.gif (2869 bytes)

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

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