Solve x^2 + 3x - 4 = 0 by completing the square

Step by step

The question

Question: solve x² + 3x − 4 = 0 by completing the square.

1 Move the −4

We want only the x parts on the left.

The − 4 is in the way.

So we add 4 on both sides.

− 4 and + 4 make zero.

Cross them out.

0 plus 4 is 4.

Now we have x² + 3x = 4.

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.

4 is 16/4.

Same bottom, so we add the tops: 16 plus 9 is 25.

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 = 25/4.

4 Square root of both sides

Something squared is 25/4.

So that something is the square root of 25/4, 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.

5 times 5 is 25.

So x + 3/2 is plus or minus that.

5 Get x alone

To remove the + 3/2, we take away 3/2 on both sides.

We do it in both equations.

Same bottom, so we work with the tops.

5/2 minus 3/2 is 1.

Minus 5/2 minus 3/2 is −4.

So x = 1 or x = −4.

6 Check

Let's check: put x = 1 into the left side.

We get 0, the same as the right side.

It works!

Now put x = −4 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 −4.

Then: complete the square.

Then: write it as a square.

Then: square root of both sides.

Then: get x alone.

Last: check.