Solve 3x^2 - 12x = 15 by completing the square Step by Step

Step by step

The question

Question: solve 3x² − 12x = 15 by completing the square.

3x² − 12x = 15

1 Divide by 3

We want just x², not 3x².

So we divide every part by 3.

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

3, −12, 15: all divide by 3

Now we have x² − 4x = 5.

x² − 4x = 5

2 Complete the square

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

A picture shows what is missing.

So we add 4 to fill the corner.

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

x² − 4x + 4 = 5 + 4

Careful!

Many people add 4 only on the left.

Then the two sides are not equal any more.

Careful: add 4 on BOTH sides

3 Write it as a square

The full square is x − 2 on each side.

So x² − 4x + 4 is (x − 2)².

x² − 4x + 4 = (x − 2)²

5 plus 4 is 9.

5 + 4 = 9

Now we have (x − 2)² = 9.

(x − 2)² = 9

4 Square root of both sides

Something squared is 9.

So that something is the square root of 9, or minus it.

Both work, because minus times minus is plus.

( )² = k → ( ) = √k or −√k

The square root of 9 is 3, because 3 times 3 is 9.

x − 2 = 3 or x − 2 = −3

5 Get x alone

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

We do it in both equations.

x = 3 + 2 or x = −3 + 2

3 plus 2 is 5.

Minus 3 plus 2 is −1.

3 + 2 = 5, −3 + 2 = −1

So x = 5 or x = −1.

x = 5 or x = −1

6 Check

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

x = 5: 3(5)² − 12(5) = 75 − 60 = 15

We get 15, the same as the right side.

It works!

Now put x = −1 into the left side.

x = −1: 3(−1)² − 12(−1) = 3 + 12 = 15

We get 15, the same as the right side.

So both answers are right!

Let's look back.

First: divide by 3.

Then: complete the square.

Then: write it as a square.

Then: square root of both sides.

Then: get x alone.

Last: check.

Answer