Step by step
The question
Question: solve 2x² − 2x − 24 = 0 by completing the square.
1 Divide by 2
We want just x², not 2x².
So we divide every part by 2.
We do it on both sides, so they stay equal.
Some numbers become fractions.
That is fine.
Now we have x² − x − 12 = 0.
2 Move the −12
We want only the x parts on the left.
The − 12 is in the way.
So we add 12 on both sides.
− 12 and + 12 make zero.
Cross them out.
0 plus 12 is 12.
Now we have x² − x = 12.
3 Complete the square
We want the left side to be one bracket, squared.
A picture shows what is missing.
So we add 1/4 to fill the corner.
We do it on both sides, so they stay equal.
4 Write it as a square
The full square is x − 1/2 on each side.
So the left side is (x − 1/2)².
To add, both need the same bottom: 4.
12 is 48/4.
Same bottom, so we add the tops: 48 plus 1 is 49.
Careful!
Many people add 1/4 only on the left.
Then the two sides are not equal any more.
Now we have (x − 1/2)² = 49/4.
5 Square root of both sides
Something squared is 49/4.
So that something is the square root of 49/4, 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.
7 times 7 is 49.
So x − 1/2 is plus or minus that.
6 Get x alone
To remove the − 1/2, we add 1/2 on both sides.
We do it in both equations.
Same bottom, so we work with the tops.
7/2 plus 1/2 is 4.
Minus 7/2 plus 1/2 is −3.
So x = 4 or x = −3.
7 Check
Let's check: put x = 4 into the left side.
We get 0, the same as the right side.
It works!
Now put x = −3 into the left side.
We get 0, the same as the right side.
So both answers are right!
Let's look back.
First: divide by 2.
Then: move the −12.
Then: complete the square.
Then: write it as a square.
Then: square root of both sides.
Then: get x alone.
Last: check.