Solve x^2 - 7x + 10 = 0 by completing the square

Step by step

The question

Question: solve x² − 7x + 10 = 0 by completing the square.

1 Move the 10

We want only the x parts on the left.

The + 10 is in the way.

So we take away 10 on both sides.

+ 10 and − 10 make zero.

Cross them out.

0 minus 10 is −10.

Now we have x² − 7x = −10.

2 Complete the square

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

A picture shows what is missing.

So we add 49/4 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 − 7/2 on each side.

So the left side is (x − 7/2)².

To add, both need the same bottom: 4.

−10 is −40/4.

Same bottom, so we add the tops: −40 plus 49 is 9.

Careful!

Many people add 49/4 only on the left.

Then the two sides are not equal any more.

Now we have (x − 7/2)² = 9/4.

4 Square root of both sides

Something squared is 9/4.

So that something is the square root of 9/4, 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.

3 times 3 is 9.

So x − 7/2 is plus or minus that.

5 Get x alone

To remove the − 7/2, we add 7/2 on both sides.

We do it in both equations.

Same bottom, so we work with the tops.

3/2 plus 7/2 is 5.

Minus 3/2 plus 7/2 is 2.

So x = 5 or x = 2.

6 Check

Let's check: put x = 5 into the left side.

We get 0, the same as the right side.

It works!

Now put x = 2 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 10.

Then: complete the square.

Then: write it as a square.

Then: square root of both sides.

Then: get x alone.

Last: check.