Solve x^2 + 6x + 2 = 0 by completing the square

Step by step

The question

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

1 Move the 2

We want only the x parts on the left.

The + 2 is in the way.

So we take away 2 on both sides.

+ 2 and − 2 make zero.

Cross them out.

0 minus 2 is −2.

Now we have x² + 6x = −2.

2 Complete the square

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

A picture shows what is missing.

So we add 9 to fill the corner.

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

Careful!

Many people add 9 only on the left.

Then the two sides are not equal any more.

3 Write it as a square

The full square is x + 3 on each side.

So x² + 6x + 9 is (x + 3)².

−2 plus 9 is 7.

Now we have (x + 3)² = 7.

4 Square root of both sides

Something squared is 7.

So that something is the square root of 7, or minus it.

Both work, because minus times minus is plus.

7 is not a square number, so we keep √7.

5 Get x alone

To remove the + 3, we take away 3 on both sides.

We do it in both equations.

We write the plain number first.

So x = −3 + √7 or x = −3 − √7.

With a calculator, √7 is about 2.65.

So x is about −0.35 or about −5.65.

Let's look back.

First: move the 2.

Then: complete the square.

Then: write it as a square.

Then: square root of both sides.

Last: get x alone.