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.

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} - 5*x* - 14 = 0

*x*
^{2} - 5*x* - 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} + 6*x* + 5 = 0

*x*
^{2} + 6*x* + 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*
:
2*x*
^{2} - 16*x* + 24 = 0

2*x*
^{2} -16*x* + 24 = 2(*x*
^{2} - 8*x* + 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} + 6*x* + 9 = 0

*x*
^{2} +6*x* + 9 = (*x* + 3)(*x* + 3) = (*x* + 3)^{2} = 0

*x* + 3 = 0

*x* = - 3

Thus, the solution set is
{ -3}
.