Quadratic Equation: by Factoring

How to solve a quadratic equation by factoring: 4 examples and their solutions.

Example 1

Example

Solution

Factor the left side:
x2 - 3x = x(x - 3).

Common Monomial Factor

x(x - 3) = 0
Then either x or (x - 3) is 0.
So check these two cases.

Case 1) x = 0
x = 0 is the answer for case 1.

Case 2) x - 3 = 0
Then x = 3.
This is the answer for case 2.

Case 1) x = 0
Case 2) x = 3

Then x = 0, 3.
This means x = 0 or x = 3.

So x = 0, 3.

Example 2

Example

Solution

Factor the left side
x2 + 5x - 14.

Find a pair of numbers
whose product is the constant term -14
and whose sum is the coefficient of the middle term +5.

-2⋅7 = -14
-2 + 7 = +5

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

Factor a Quadratic Trinomial

(x - 2)(x + 7) = 0
Then either (x - 2) or (x + 7) is 0.
So check these two cases.

Case 1) x - 2 = 0
Then x = 2.
This is the answer for case 1.

Case 2) x + 7 = 0
Then x = -7.
This is the answer for case 2.

Case 1) x = 2
Case 2) x = -7

Then x = -7, 2.

So x = -7, 2.

Example 3

Example

Solution

Change the left side
to a perfect square trinomial.

x2 is x2.

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

+36 is +62.

x2 + 2⋅x⋅6 + 62 = (x + 6)2.

Factor a Perfect Square Trinomial

(x + 6)2 = 0

Square root both sides.

Then x + 6 = 0.
(x + 6 = ±0 = 0)

Quadratic Equation: Square Root

Move +6 to the right side.

Then x = -6.

So x = -6.

Example 4

Example

Solution

You can move -81 to the right side
and square root it.
But let's solve this example differently.

-81 = -92

x2 - 92 = (x + 9)(x - 9)

Factor the Difference of Two Squares: a2 - b2

(x + 9)(x - 9) = 0
Then either (x + 9) or (x - 9) is 0.
So check these two cases.

Case 1) x + 9 = 0
Then x = -9.
This is the answer for case 1.

Case 2) x - 9 = 0
Then x = 9.
This is the answer for case 2.

Case 1) x = -9
Case 2) x = 9

Then x = ±9.

So x = ±9.