Solve x^2 - 3x - 1 = 0 by completing the square

Step by step

The question

Question: solve x² − 3x − 1 = 0 by completing the square.

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.

− 1 and + 1 make zero.

Cross them out.

0 plus 1 is 1.

Now we have x² − 3x = 1.

2 Complete the square

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

A picture shows what is missing.

So we add 9/4 to fill the corner.

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

3 Write it as a square

The full square is x − 3/2 on each side.

So the left side is (x − 3/2)².

To add, both need the same bottom: 4.

1 is 4/4.

Same bottom, so we add the tops: 4 plus 9 is 13.

Careful!

Many people add 9/4 only on the left.

Then the two sides are not equal any more.

Now we have (x − 3/2)² = 13/4.

4 Square root of both sides

Something squared is 13/4.

So that something is the square root of 13/4, or minus it.

Both work, because minus times minus is plus.

The square root of a fraction is the root of the top over the root of the bottom.

2 times 2 is 4.

13 is not a square number, so we keep √13.

So x − 3/2 is plus or minus that.

5 Get x alone

To remove the − 3/2, we add 3/2 on both sides.

We do it in both equations.

We write the plain number first.

So x = 3/2 + √13/2 or x = 3/2 − √13/2.

With a calculator, x is about 3.3 or about −0.3.

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.