Complete the square: x^2 - 4x - 5 = 0 Step by Step

Step by step

The question

Question: solve x² − 4x − 5 = 0 by completing the square.

x² − 4x − 5 = 0

1 Move the −5

We want only the x parts on the left.

The − 5 is in the way.

So we add 5 on both sides.

x² − 4x ~~− 5 + 5~~ = 0 + 5

− 5 and + 5 make zero.

Cross them out.

0 plus 5 is 5.

0 + 5 = 5

Now we have x² − 4x = 5.

x² − 4x = 5

2 Complete the square

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

A picture shows what is missing.

So we add 4 to fill the corner.

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

x² − 4x + 4 = 5 + 4

Careful!

Many people add 4 only on the left.

Then the two sides are not equal any more.

Careful: add 4 on BOTH sides

3 Write it as a square

The full square is x − 2 on each side.

So x² − 4x + 4 is (x − 2)².

x² − 4x + 4 = (x − 2)²

5 plus 4 is 9.

5 + 4 = 9

Now we have (x − 2)² = 9.

(x − 2)² = 9

4 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 − 2 = 3 or x − 2 = −3

5 Get x alone

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

We do it in both equations.

x = 3 + 2 or x = −3 + 2

3 plus 2 is 5.

Minus 3 plus 2 is −1.

3 + 2 = 5, −3 + 2 = −1

So x = 5 or x = −1.

x = 5 or x = −1

6 Check

Let's check: put x = 5 into the left side.

x = 5: 5² − 4(5) − 5 = 25 − 20 − 5 = 0

We get 0, the same as the right side.

It works!

Now put x = −1 into the left side.

x = −1: (−1)² − 4(−1) − 5 = 1 + 4 − 5 = 0

We get 0, the same as the right side.

So both answers are right!

Let's look back.

First: move the −5.

Then: complete the square.

Then: write it as a square.

Then: square root of both sides.

Then: get x alone.

Last: check.

Answer