Solve 2x^2 - 4x - 6 = 0 by completing the square

Step by step

The question

Question: solve 2x² − 4x − 6 = 0 by completing the square.

1 Divide by 2

We want just x², not 2x².

So we divide every part by 2.

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

Now we have x² − 2x − 3 = 0.

2 Move the −3

We want only the x parts on the left.

The − 3 is in the way.

So we add 3 on both sides.

− 3 and + 3 make zero.

Cross them out.

0 plus 3 is 3.

Now we have x² − 2x = 3.

3 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.

Careful!

Many people add 1 only on the left.

Then the two sides are not equal any more.

4 Write it as a square

The full square is x − 1 on each side.

So x² − 2x + 1 is (x − 1)².

3 plus 1 is 4.

Now we have (x − 1)² = 4.

5 Square root of both sides

Something squared is 4.

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

Both work, because minus times minus is plus.

The square root of 4 is 2, because 2 times 2 is 4.

6 Get x alone

To remove the − 1, we add 1 on both sides.

We do it in both equations.

2 plus 1 is 3.

Minus 2 plus 1 is −1.

So x = 3 or x = −1.

7 Check

Let's check: put x = 3 into the left side.

We get 0, the same as the right side.

It works!

Now put x = −1 into the left side.

We get 0, the same as the right side.

So both answers are right!

Let's look back.

First: divide by 2.

Then: move the −3.

Then: complete the square.

Then: write it as a square.

Then: square root of both sides.

Then: get x alone.

Last: check.