Solve x^2 + 4x = 1 by completing the square Step by Step

Step by step

The question

Question: solve x² + 4x = 1 by completing the square.

x² + 4x = 1

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 = 1 + 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

1 plus 4 is 5.

1 + 4 = 5

Now we have (x + 2)² = 5.

(x + 2)² = 5

3 Square root of both sides

Something squared is 5.

So that something is the square root of 5, or minus it.

Both work, because minus times minus is plus.

( )² = k → ( ) = √k or −√k

5 is not a square number, so we keep √5.

x + 2 = √5 or x + 2 = −√5

4 Get x alone

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

We do it in both equations.

x = √5 − 2 or x = −√5 − 2

We write the plain number first.

So x = −2 + √5 or x = −2 − √5.

x = −2 + √5 or x = −2 − √5

With a calculator, √5 is about 2.24.

So x is about 0.24 or about −4.24.

x ≈ 0.24 or x ≈ −4.24

Let's look back.

First: complete the square.

Then: write it as a square.

Then: square root of both sides.

Last: get x alone.

Answer