Solve x^2 - 10x = -16 by completing the square Step by Step

Step by step

The question

Question: solve x² − 10x = −16 by completing the square.

x² − 10x = −16

1 Complete the square

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

A picture shows what is missing.

So we add 25 to fill the corner.

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

x² − 10x + 25 = −16 + 25

Careful!

Many people add 25 only on the left.

Then the two sides are not equal any more.

Careful: add 25 on BOTH sides

2 Write it as a square

The full square is x − 5 on each side.

So x² − 10x + 25 is (x − 5)².

x² − 10x + 25 = (x − 5)²

−16 plus 25 is 9.

−16 + 25 = 9

Now we have (x − 5)² = 9.

(x − 5)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 − 5 = 3 or x − 5 = −3

4 Get x alone

To remove the − 5, we add 5 on both sides.

We do it in both equations.

x = 3 + 5 or x = −3 + 5

3 plus 5 is 8.

Minus 3 plus 5 is 2.

3 + 5 = 8, −3 + 5 = 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 − 10(8) = 64 − 80 = −16

We get −16, the same as the right side.

It works!

Now put x = 2 into the left side.

x = 2: 2² − 10(2) = 4 − 20 = −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