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

Step by step

The question

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

x² + 3x = 0

1 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.

x² + 3x + 94 = 0 + 94

2 Write it as a square

The full square is x + 3/2 on each side.

So the left side is (x + 3/2)².

x² + 3x + 94 = (x + 32)²

To add, both need the same bottom: 4.

0 is 0/4.

Same bottom, so we add the tops: 0 plus 9 is 9.

0 = 04

04 + 94 = 94

Careful!

Many people add 9/4 only on the left.

Then the two sides are not equal any more.

Careful: add 94 on BOTH sides

Now we have (x + 3/2)² = 9/4.

(x + 32)² = 94

3 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.

( )² = k → ( ) = √k or −√k

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.

√94 = √9√4 = 32

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

x + 32 = 32 or x + 32 = −32

4 Get x alone

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

We do it in both equations.

x = 32 − 32 or x = −32 − 32

Same bottom, so we work with the tops.

3/2 minus 3/2 is 0.

Minus 3/2 minus 3/2 is −3.

32 − 32 = 02 = 0

−32 − 32 = −62 = −3

So x = 0 or x = −3.

x = 0 or x = −3

5 Check

Let's check: put x = 0 into the left side.

x = 0: 0² + 3(0) = 0 + 0 = 0

We get 0, the same as the right side.

It works!

Now put x = −3 into the left side.

x = −3: (−3)² + 3(−3) = 9 − 9 = 0

We get 0, the same as the right side.

So both answers are right!

Let's look back.

First: complete the square.

Then: write it as a square.

Then: square root of both sides.

Then: get x alone.

Last: check.

Answer