Solve x^2 + 2x - 1 = 0 by completing the square Step by Step

Step by step

The question

Question: solve x² + 2x − 1 = 0 by completing the square.

x² + 2x − 1 = 0

1 Move the −1

We want only the x parts on the left.

The − 1 is in the way.

So we add 1 on both sides.

x² + 2x ~~− 1 + 1~~ = 0 + 1

− 1 and + 1 make zero.

Cross them out.

0 plus 1 is 1.

0 + 1 = 1

Now we have x² + 2x = 1.

x² + 2x = 1

2 Complete the square

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

A picture shows what is missing.

So we add 1 to fill the corner.

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

x² + 2x + 1 = 1 + 1

Careful!

Many people add 1 only on the left.

Then the two sides are not equal any more.

Careful: add 1 on BOTH sides

3 Write it as a square

The full square is x + 1 on each side.

So x² + 2x + 1 is (x + 1)².

x² + 2x + 1 = (x + 1)²

1 plus 1 is 2.

1 + 1 = 2

Now we have (x + 1)² = 2.

(x + 1)² = 2

4 Square root of both sides

Something squared is 2.

So that something is the square root of 2, or minus it.

Both work, because minus times minus is plus.

( )² = k → ( ) = √k or −√k

2 is not a square number, so we keep √2.

x + 1 = √2 or x + 1 = −√2

5 Get x alone

To remove the + 1, we take away 1 on both sides.

We do it in both equations.

x = √2 − 1 or x = −√2 − 1

We write the plain number first.

So x = −1 + √2 or x = −1 − √2.

x = −1 + √2 or x = −1 − √2

With a calculator, √2 is about 1.41.

So x is about 0.41 or about −2.41.

x ≈ 0.41 or x ≈ −2.41

Let's look back.

First: move the −1.

Then: complete the square.

Then: write it as a square.

Then: square root of both sides.

Last: get x alone.

Answer