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 x
^{2} + 3x + 2 = 0
First, you have to factor x
^{2} + 3x + 2
Since the coefficient of x
^{2} is 1 (x
^{2} =
1x
^{2}), 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, x
^{2} + 3x + 2 = ( x + 2) × ( x + 1)
x
^{2} + 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 x
^{2} + 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 x
^{2} + -3x + 2 = 0, you will do:
x
^{2} + -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 x
^{2} + x -30 = 0
First, you have to factor x
^{2} + 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), x
^{2} + x -30 = ( x + 6) × ( x - 5)
x
^{2} + 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 x
^{2} + 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 x
^{2} + -x -30 = 0, you will do:
x
^{2} + -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
6x
^{2} + 27x + 30 = 0
First factor 6x
^{2} + 27x + 30
6x
^{2} + 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!
6x
^{2} + 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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