# 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:

x^{2} - 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

x^{2} + 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.

x^{2} is x^{2}.

+12x is

+2 times

x times,

(+12x)/(+2⋅x), 6.

+36 is +6^{2}.

x^{2} + 2⋅x⋅6 + 6^{2} = (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 = -9^{2}

x^{2} - 9^{2} = (x + 9)(x - 9)

Factor the Difference of Two Squares: a^{2} - b^{2}

(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.