Solve x^2 - 9x = -18 by completing the square Step by Step

Step by step

The question

Question: solve x² − 9x = −18 by completing the square.

x² − 9x = −18

1 Complete the square

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

A picture shows what is missing.

So we add 81/4 to fill the corner.

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

x² − 9x + 814 = −18 + 814

2 Write it as a square

The full square is x − 9/2 on each side.

So the left side is (x − 9/2)².

x² − 9x + 814 = (x − 92)²

To add, both need the same bottom: 4.

−18 is −72/4.

Same bottom, so we add the tops: −72 plus 81 is 9.

−18 = −724

−724 + 814 = 94

Careful!

Many people add 81/4 only on the left.

Then the two sides are not equal any more.

Careful: add 814 on BOTH sides

Now we have (x − 9/2)² = 9/4.

(x − 92)² = 94

3 Square root of both sides

Something squared is 9/4.

So that something is the square root of 9/4, 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.

3 times 3 is 9.

√94 = √9√4 = 32

So x − 9/2 is plus or minus that.

x − 92 = 32 or x − 92 = −32

4 Get x alone

To remove the − 9/2, we add 9/2 on both sides.

We do it in both equations.

x = 32 + 92 or x = −32 + 92

Same bottom, so we work with the tops.

3/2 plus 9/2 is 6.

Minus 3/2 plus 9/2 is 3.

32 + 92 = 122 = 6

−32 + 92 = 62 = 3

So x = 6 or x = 3.

x = 6 or x = 3

5 Check

Let's check: put x = 6 into the left side.

x = 6: 6² − 9(6) = 36 − 54 = −18

We get −18, the same as the right side.

It works!

Now put x = 3 into the left side.

x = 3: 3² − 9(3) = 9 − 27 = −18

We get −18, 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