Solve x^2 + 7x = -12 by completing the square Step by Step

Step by step

The question

Question: solve x² + 7x = −12 by completing the square.

x² + 7x = −12

1 Complete the square

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

A picture shows what is missing.

So we add 49/4 to fill the corner.

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

x² + 7x + 494 = −12 + 494

2 Write it as a square

The full square is x + 7/2 on each side.

So the left side is (x + 7/2)².

x² + 7x + 494 = (x + 72)²

To add, both need the same bottom: 4.

−12 is −48/4.

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

−12 = −484

−484 + 494 = 14

Careful!

Many people add 49/4 only on the left.

Then the two sides are not equal any more.

Careful: add 494 on BOTH sides

Now we have (x + 7/2)² = 1/4.

(x + 72)² = 14

3 Square root of both sides

Something squared is 1/4.

So that something is the square root of 1/4, 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.

1 times 1 is 1.

√14 = √1√4 = 12

So x + 7/2 is plus or minus that.

x + 72 = 12 or x + 72 = −12

4 Get x alone

To remove the + 7/2, we take away 7/2 on both sides.

We do it in both equations.

x = 12 − 72 or x = −12 − 72

Same bottom, so we work with the tops.

1/2 minus 7/2 is −3.

Minus 1/2 minus 7/2 is −4.

12 − 72 = −62 = −3

−12 − 72 = −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: (−3)² + 7(−3) = 9 − 21 = −12

We get −12, the same as the right side.

It works!

Now put x = −4 into the left side.

x = −4: (−4)² + 7(−4) = 16 − 28 = −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