Solve x^2 + 2x = 8 by completing the square Step by Step

Step by step

The question

Question: solve x² + 2x = 8 by completing the square.

x² + 2x = 8

1 Complete the square

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

A picture shows what is missing.

So we add 1 to fill the corner.

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

x² + 2x + 1 = 8 + 1

Careful!

Many people add 1 only on the left.

Then the two sides are not equal any more.

Careful: add 1 on BOTH sides

2 Write it as a square

The full square is x + 1 on each side.

So x² + 2x + 1 is (x + 1)².

x² + 2x + 1 = (x + 1)²

8 plus 1 is 9.

8 + 1 = 9

Now we have (x + 1)² = 9.

(x + 1)² = 9

3 Square root of both sides

Something squared is 9.

So that something is the square root of 9, or minus it.

Both work, because minus times minus is plus.

( )² = k → ( ) = √k or −√k

The square root of 9 is 3, because 3 times 3 is 9.

x + 1 = 3 or x + 1 = −3

4 Get x alone

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

We do it in both equations.

x = 3 − 1 or x = −3 − 1

3 minus 1 is 2.

Minus 3 minus 1 is −4.

3 − 1 = 2, −3 − 1 = −4

So x = 2 or x = −4.

x = 2 or x = −4

5 Check

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

x = 2: 2² + 2(2) = 4 + 4 = 8

We get 8, the same as the right side.

It works!

Now put x = −4 into the left side.

x = −4: (−4)² + 2(−4) = 16 − 8 = 8

We get 8, 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