Solve x^2 + 8x = 9 by completing the square Step by Step

Step by step

The question

Question: solve x² + 8x = 9 by completing the square.

x² + 8x = 9

1 Complete the square

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

A picture shows what is missing.

So we add 16 to fill the corner.

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

x² + 8x + 16 = 9 + 16

Careful!

Many people add 16 only on the left.

Then the two sides are not equal any more.

Careful: add 16 on BOTH sides

2 Write it as a square

The full square is x + 4 on each side.

So x² + 8x + 16 is (x + 4)².

x² + 8x + 16 = (x + 4)²

9 plus 16 is 25.

9 + 16 = 25

Now we have (x + 4)² = 25.

(x + 4)² = 25

3 Square root of both sides

Something squared is 25.

So that something is the square root of 25, or minus it.

Both work, because minus times minus is plus.

( )² = k → ( ) = √k or −√k

The square root of 25 is 5, because 5 times 5 is 25.

x + 4 = 5 or x + 4 = −5

4 Get x alone

To remove the + 4, we take away 4 on both sides.

We do it in both equations.

x = 5 − 4 or x = −5 − 4

5 minus 4 is 1.

Minus 5 minus 4 is −9.

5 − 4 = 1, −5 − 4 = −9

So x = 1 or x = −9.

x = 1 or x = −9

5 Check

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

x = 1: 1² + 8(1) = 1 + 8 = 9

We get 9, the same as the right side.

It works!

Now put x = −9 into the left side.

x = −9: (−9)² + 8(−9) = 81 − 72 = 9

We get 9, 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