Solving quadratic equations by factoring


Solving quadratic equations by factoring could be many times the simplest and quickest way to solve quadratic equation as long as you know how to factor.

I strongly recommend you to study or review the following important unit: Factoring

Example #1: Solving quadratic equations by factoring

Solve x2 + 3x + 2 = 0

First, you have to factor x2 + 3x + 2

Since the coefficient of x2 is 1 (x2 = 1x2), you can factor by looking for factors of the last term (last term is 2) that add up to the coefficient of the second term (3x, coefficient is 3)

2 = 1 × 2

2 = -1 × -2

Since 1 + 2 = 3, and 3 is the coefficient of the second term, x2 + 3x + 2 = ( x + 2) × ( x + 1)

x2 + 3x + 2 = 0 gives:

( x + 2) × ( x + 1) = 0

( x + 2) × ( x + 1) = 0 when either x + 2 = 0 or x + 1 = 0

x + 2 = 0 when x = -2

x + 1 = 0 when x = -1

Let us now check x = -2 and x = -1 are indeed solutions of x2 + 3x + 2 = 0

(-2)2 + 3 × -2 + 2 = 4 + -6 + 2 = 3 + -6 = 0

(-1)2 + 3 × -1 + 2 = 1 + -3 + 2 = 3 + -3 = 0


If instead you were solving x2 + -3x + 2 = 0, you will do:

x2 + -3x + 2 = 0


( x + -2) × ( x + -1) = 0

( x + -2) × ( x + -1) = 0 when either x + -2 = 0 or x + -1 = 0

x + -2 = 0 when x = 2

x + -1 = 0 when x = 1

Check that this are indeed the solutions

Example #2: Solving quadratic equations by factoring

Solve x2 + x -30 = 0

First, you have to factor x2 + x -30

-30 = 30 × -1

-30 = 15 × -2

-30 = 6 × -5

-30 = -30 × 1

-30 = -15 × 2

-30 = -6 × 5

Since only 6 + -5 = 1, and 1 is the coefficient of the second term(x = 1x), x2 + x -30 = ( x + 6) × ( x - 5)

x2 + x -30 = 0 gives:

( x + 6) × ( x - 5) = 0

( x + 6) × ( x - 5) = 0 when either x + 6 = 0 or x - 5 = 0

x + 6 = 0 when x = -6

x - 5 = 0 when x = 5

Let us now check x = -6 and x = 5 are indeed solutions of x2 + x -30 = 0

(-6)2 + -6 -30 = 36 - 6 - 30 = 30 - 30 = 0

(5)2 + 5 - 30 = 25 + 5 - 30 = 30 - 30 = 0


If instead you were solving x2 + -x -30 = 0, you will do:

x2 + -x - 30 = 0


( x - 6) × ( x + 5) = 0

( x - 6) × ( x + 5) = 0 when either x - 6 = 0 or x + 5 = 0

x - 6 = 0 when x = 6

x + 5 = 0 when x = -5

Check that this are indeed the solutions

Solving quadratic equations by factoring can get very tough. See below:

Example #3: Solving quadratic equations by factoring

6x2 + 27x + 30 = 0

First factor 6x2 + 27x + 30

6x2 + 27x + 30 = ( 3x + ?) × (2x + ?) or ( 6x + ?) × (x + ?)

Now, factor the last term 30

30 = 30 × 1

30 = 15 × 2

30 = 6 × 5

To get the 27x, you have to try out the cross multiplications below. There are 6 of them. Cross multiply and add!

6x         x
30         1

6x + 30x = 36x

6x         x
15         2

12x + 15x = 27x

6x         x
6          5

30x + 6x = 36x

3x         2x
30         1

3x + 60x = 63x

3x         2x
15         2

6x + 30x = 36x

3x         2x
6           5

15x + 12x = 27x

You got a couple of choices shown in bold!

6x2 + 27x + 30 = 0

(6x + 15) × ( x + 2)= 0

6x + 15 = 0

6x = -15

x = -15/6

x + 2 = 0

x = -2




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