Solving quadratic equations by factoring

Solving quadratic equations by factoring could be many times the simplest and quickest way to solve quadratic equations as long as you know how to factor. I strongly recommend that you study or review the following important unit about factoring.

Solving quadratic equations by factoring

Example #1:  It explains in more details how to solve x2 + 3x + 2 = 0 or the example in the figure above.

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

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 use -1 and -2 since -1 + - 2 add up to -3.

( 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 1 and 2 are indeed the solutions

Tougher examples of solving quadratic equations by factoring.


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 use -6 and 5 since -6 + 5 = -1

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 6 and -5 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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