Complete the square: x^2 + 10x + 21 = 0

Step by step

The question

Question: solve x² + 10x + 21 = 0 by completing the square.

1 Move the 21

We want only the x parts on the left.

The + 21 is in the way.

So we take away 21 on both sides.

+ 21 and − 21 make zero.

Cross them out.

0 minus 21 is −21.

Now we have x² + 10x = −21.

2 Complete the square

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

A picture shows what is missing.

So we add 25 to fill the corner.

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

Careful!

Many people add 25 only on the left.

Then the two sides are not equal any more.

3 Write it as a square

The full square is x + 5 on each side.

So x² + 10x + 25 is (x + 5)².

−21 plus 25 is 4.

Now we have (x + 5)² = 4.

4 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.

5 Get x alone

To remove the + 5, we take away 5 on both sides.

We do it in both equations.

2 minus 5 is −3.

Minus 2 minus 5 is −7.

So x = −3 or x = −7.

6 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 = −7 into the left side.

We get 0, the same as the right side.

So both answers are right!

Let's look back.

First: move the 21.

Then: complete the square.

Then: write it as a square.

Then: square root of both sides.

Then: get x alone.

Last: check.