Solve x^2 + 6x = 7 by completing the square Step by Step

Step by step

The question

Question: solve x² + 6x = 7 by completing the square.

x² + 6x = 7

1 Complete the square

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

A picture shows what is missing.

So we add 9 to fill the corner.

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

x² + 6x + 9 = 7 + 9

Careful!

Many people add 9 only on the left.

Then the two sides are not equal any more.

Careful: add 9 on BOTH sides

2 Write it as a square

The full square is x + 3 on each side.

So x² + 6x + 9 is (x + 3)².

x² + 6x + 9 = (x + 3)²

7 plus 9 is 16.

7 + 9 = 16

Now we have (x + 3)² = 16.

(x + 3)² = 16

3 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 + 3 = 4 or x + 3 = −4

4 Get x alone

To remove the + 3, we take away 3 on both sides.

We do it in both equations.

x = 4 − 3 or x = −4 − 3

4 minus 3 is 1.

Minus 4 minus 3 is −7.

4 − 3 = 1, −4 − 3 = −7

So x = 1 or x = −7.

x = 1 or x = −7

5 Check

Let's check: put x = 1 into the left side.

x = 1: 1² + 6(1) = 1 + 6 = 7

We get 7, the same as the right side.

It works!

Now put x = −7 into the left side.

x = −7: (−7)² + 6(−7) = 49 − 42 = 7

We get 7, 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