Quadratics
Factoring Quadratic Equations
A quadratic equation is an equation of the form ax 2 + bx + c = 0 , where a≠ 0 , and a , b , and c are real numbers.
Solving Quadratic Equations by Factoring
We can often factor a quadratic equation into the product of two binomials. We are then left with an equation of the form (x + d )(x + e) = 0 , where d and e are integers.
The zero product property states that, if the product of two quantities is equal to 0 , then at least one of the quantities must be equal to zero. Thus, if (x + d )(x + e) = 0 , either (x + d )= 0 or (x + e) = 0 . Consequently, the two solutions to the equation are x = - d and x = - e .
Example 1: Solve for
x
:
x
2 - 5x - 14 = 0
x
2 - 5x - 14 = (x - 7)(x + 2) = 0
x - 7 = 0
or
x + 2 = 0
x = 7
or
x = - 2
Thus, the solution set is
{ -2, 7}
.
Example 2: Solve for
x
:
x
2 + 6x + 5 = 0
x
2 + 6x + 5 = (x + 1)(x + 5) = 0
x + 1 = 0
or
x + 5 = 0
x = - 1
or
x = - 5
Thus, the solution set is
{ -1, -5}
.
Example 3: Solve for
x
:
2x
2 - 16x + 24 = 0
2x
2 -16x + 24 = 2(x
2 - 8x + 12) = 2(x - 2)(x - 6) = 0
x - 2 = 0
or
x - 6 = 0
x = 2
or
x = 6
Thus, the solution set is
{2, 6}
.
Example 4: Solve for
x
:
x
2 + 6x + 9 = 0
x
2 +6x + 9 = (x + 3)(x + 3) = (x + 3)2 = 0
x + 3 = 0
x = - 3
Thus, the solution set is
{ -3}
.
