Solve x^2 - 6x = 16 by completing the square Step by Step

Step by step

The question

Question: solve x² − 6x = 16 by completing the square.

x² − 6x = 16

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 = 16 + 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

16 plus 9 is 25.

16 + 9 = 25

Now we have (x − 3)² = 25.

(x − 3)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 − 3 = 5 or x − 3 = −5

4 Get x alone

To remove the − 3, we add 3 on both sides.

We do it in both equations.

x = 5 + 3 or x = −5 + 3

5 plus 3 is 8.

Minus 5 plus 3 is −2.

5 + 3 = 8, −5 + 3 = −2

So x = 8 or x = −2.

x = 8 or x = −2

5 Check

Let's check: put x = 8 into the left side.

x = 8: 8² − 6(8) = 64 − 48 = 16

We get 16, the same as the right side.

It works!

Now put x = −2 into the left side.

x = −2: (−2)² − 6(−2) = 4 + 12 = 16

We get 16, 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