Solve x^2 + 12x + 20 = 0 by completing the square Step by Step

Step by step

The question

Question: solve x² + 12x + 20 = 0 by completing the square.

x² + 12x + 20 = 0

1 Move the 20

We want only the x parts on the left.

The + 20 is in the way.

So we take away 20 on both sides.

x² + 12x ~~+ 20 − 20~~ = 0 − 20

+ 20 and − 20 make zero.

Cross them out.

0 minus 20 is −20.

0 − 20 = −20

Now we have x² + 12x = −20.

x² + 12x = −20

2 Complete the square

We want the left side to be one bracket, squared.

A picture shows what is missing.

So we add 36 to fill the corner.

We do it on both sides, so they stay equal.

x² + 12x + 36 = −20 + 36

Careful!

Many people add 36 only on the left.

Then the two sides are not equal any more.

Careful: add 36 on BOTH sides

3 Write it as a square

The full square is x + 6 on each side.

So x² + 12x + 36 is (x + 6)².

x² + 12x + 36 = (x + 6)²

−20 plus 36 is 16.

−20 + 36 = 16

Now we have (x + 6)² = 16.

(x + 6)² = 16

4 Square root of both sides

Something squared is 16.

So that something is the square root of 16, or minus it.

Both work, because minus times minus is plus.

( )² = k → ( ) = √k or −√k

The square root of 16 is 4, because 4 times 4 is 16.

x + 6 = 4 or x + 6 = −4

5 Get x alone

To remove the + 6, we take away 6 on both sides.

We do it in both equations.

x = 4 − 6 or x = −4 − 6

4 minus 6 is −2.

Minus 4 minus 6 is −10.

4 − 6 = −2, −4 − 6 = −10

So x = −2 or x = −10.

x = −2 or x = −10

6 Check

Let's check: put x = −2 into the left side.

x = −2: (−2)² + 12(−2) + 20 = 4 − 24 + 20

= 0

We get 0, the same as the right side.

It works!

Now put x = −10 into the left side.

x = −10: (−10)² + 12(−10) + 20

= 100 − 120 + 20 = 0

We get 0, the same as the right side.

So both answers are right!

Let's look back.

First: move the 20.

Then: complete the square.

Then: write it as a square.

Then: square root of both sides.

Then: get x alone.

Last: check.

Answer