Solve x^2 + 4x = 5 by completing the square Step by Step

Step by step

The question

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

x² + 4x = 5

1 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

2 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

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 + 2 = 3 or x + 2 = −3

4 Get x alone

To remove the + 2, we take away 2 on both sides.

We do it in both equations.

x = 3 − 2 or x = −3 − 2

3 minus 2 is 1.

Minus 3 minus 2 is −5.

3 − 2 = 1, −3 − 2 = −5

So x = 1 or x = −5.

x = 1 or x = −5

5 Check

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

x = 1: 1² + 4(1) = 1 + 4 = 5

We get 5, the same as the right side.

It works!

Now put x = −5 into the left side.

x = −5: (−5)² + 4(−5) = 25 − 20 = 5

We get 5, 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