Solving a Quadratic Equation by Factoring

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.

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

Solve the given equation. x^2 - 3x = 0

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

Solve the given equation. x^2 + 5x - 14 = 0

Factor the left side:
x2 + 5x - 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

Solve the given equation. x^2 + 12x + 36 = 0

See if the left side is a perfect square trinomial.

x2 is x2.

+12x is
+2 times
x times,
(+12x)/(+2⋅x), 6.

And +36 is 62.

x2 + 2⋅x⋅6 + 62
is a perfect square trinomial.

So factor x2 + 2⋅x⋅6 + 62:
(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

Solve the given equation. x^2 - 81 = 0

See if the left side is the difference of squares.

x2 is x2.

And 81 is 92.

x2 - 92
is the difference of squares.

So factor x2 - 92:
(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

Solve the given equation. x^2 - 81 = 0

You can also solve this quadratic equation
by square rooting.

First move -81 to the right side.
Then x2 = 81.

Square root both sides.

Then x = ±√81.

Don't forget to write ± sign,
because (-√81)2 can also be 81.

81 = 92

So ±√81 = ±√92.

Take the squared factor [9]
out from the square root sign.

Then x = ±9.

As you can see,
you got the same answer.