# Solving a Quadratic Equation by Factoring

How to solve a quadratic equation by factoring: zero product property, examples and their solutions.

## Zero Product Property

If a product of two factors is 0,

then either one of the factor is 0.

You're going to use this property

to solve quadratic equations.

## Example 1

Factor the left side:*x*(*x* - 3) = 0.

Factoring - Using the distributive property

*x*(*x* - 3) = 0

So either [*x*] or (*x* - 3) is 0.

1) *x* = 0

2) *x* - 3 = 0

So *x* = 3.

So *x* is 0 or 3.

You can just write *x* = 0, 3.

## Example 2

Factor the left side:*x*^{2} + 5*x* - 14.

Factoring a quadratic trinomial

Find the pair of numbers

whose product is the constant term [-14]

and whose sum is the middle term's coefficient [+5].

The constant term is (-).

So the signs of the numbers are different:

one is (+), and the other is (-).

[-14] = -1⋅14

-1 + 14 = 13 ≠ [+5]

So -1 and 14 are not the right numbers.

[-14] = -2⋅7

-2 + 7 = 5 = [+5]

So -2 and 7 are the right numbers.

Use -2 and +7

to write a factored form:

(*x* - 2)(*x* + 7) = 0.

(*x* - 2)(*x* + 7) = 0

So either (*x* - 2) or (*x* + 7) is 0.

1) *x* - 2 = 0

So *x* = 2.

2) *x* + 7 = 0

So *x* = -7.

So *x* = -7, 2.

## Example 3

See if the left side is a perfect square trinomial.*x*^{2} is *x*^{2}.

+12*x* is

+2 times*x* times,

(+12*x*)/(+2⋅*x*), 6.

And +36 is 6^{2}.

*x*^{2} + 2⋅*x*⋅6 + 6^{2}

is a perfect square trinomial.

So factor *x*^{2} + 2⋅*x*⋅6 + 6^{2}:

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

Factoring a perfect square trinomial

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

So *x* + 6 = 0.

Move +6 to the right side.

Then *x* = -6.

## Example 4

See if the left side is the difference of squares.*x*^{2} is *x*^{2}.

And 81 is 9^{2}.

*x*^{2} - 9^{2}

is the difference of squares.

So factor *x*^{2} - 9^{2}:

(*x* + 9)(*x* - 9) = 0.

Factoring the difference of squares

(*x* + 9)(*x* - 9) = 0

So either (*x* + 9) or (*x* - 9) is 0.

1) *x* + 9 = 0

So *x* = -9.

2) *x* - 9 = 0

So *x* = 9.

So *x* = -9, 9.

Or you can combine these two: *x* = ±9.

## Example 4: Another Solution

You can also solve this quadratic equation

by square rooting.

First move -81 to the right side.

Then *x*^{2} = 81.

Square root both sides.

Then *x* = ±√81.

Don't forget to write ± sign,

because (-√81)^{2} can also be 81.

81 = 9^{2}

So ±√81 = ±√9^{2}.

Take the squared factor [9]

out from the square root sign.

Then *x* = ±9.

As you can see,

you got the same answer.