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

Step by step

The question

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

x² + x = 12

1 Complete the square

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

A picture shows what is missing.

So we add 1/4 to fill the corner.

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

x² + x + 14 = 12 + 14

2 Write it as a square

The full square is x + 1/2 on each side.

So the left side is (x + 1/2)².

x² + x + 14 = (x + 12)²

To add, both need the same bottom: 4.

12 is 48/4.

Same bottom, so we add the tops: 48 plus 1 is 49.

12 = 484

484 + 14 = 494

Careful!

Many people add 1/4 only on the left.

Then the two sides are not equal any more.

Careful: add 14 on BOTH sides

Now we have (x + 1/2)² = 49/4.

(x + 12)² = 494

3 Square root of both sides

Something squared is 49/4.

So that something is the square root of 49/4, or minus it.

Both work, because minus times minus is plus.

( )² = k → ( ) = √k or −√k

The square root of a fraction is the root of the top over the root of the bottom.

2 times 2 is 4.

7 times 7 is 49.

√494 = √49√4 = 72

So x + 1/2 is plus or minus that.

x + 12 = 72 or x + 12 = −72

4 Get x alone

To remove the + 1/2, we take away 1/2 on both sides.

We do it in both equations.

x = 72 − 12 or x = −72 − 12

Same bottom, so we work with the tops.

7/2 minus 1/2 is 3.

Minus 7/2 minus 1/2 is −4.

72 − 12 = 62 = 3

−72 − 12 = −82 = −4

So x = 3 or x = −4.

x = 3 or x = −4

5 Check

Let's check: put x = 3 into the left side.

x = 3: 3² + 3 = 9 + 3 = 12

We get 12, the same as the right side.

It works!

Now put x = −4 into the left side.

x = −4: (−4)² + (−4) = 16 − 4 = 12

We get 12, the same as the right side.

So both answers are right!

Let's look back.

First: complete the square.

Then: write it as a square.

Then: square root of both sides.

Then: get x alone.

Last: check.

Answer