Step by step
The question
Question: solve 2x² − 3x = 20 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 − 3x/2 = 10.
2 Complete the square
We want the left side to be one bracket, squared.
A picture shows what is missing.
So we add 9/16 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 − 3/4 on each side.
So the left side is (x − 3/4)².
To add, both need the same bottom: 16.
10 is 160/16.
Same bottom, so we add the tops: 160 plus 9 is 169.
Careful!
Many people add 9/16 only on the left.
Then the two sides are not equal any more.
Now we have (x − 3/4)² = 169/16.
4 Square root of both sides
Something squared is 169/16.
So that something is the square root of 169/16, 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.
4 times 4 is 16.
13 times 13 is 169.
So x − 3/4 is plus or minus that.
5 Get x alone
To remove the − 3/4, we add 3/4 on both sides.
We do it in both equations.
Same bottom, so we work with the tops.
13/4 plus 3/4 is 4.
Minus 13/4 plus 3/4 is −5/2.
So x = 4 or x = −5/2.
6 Check
Let's check: put x = 4 into the left side.
We get 20, the same as the right side.
It works!
Now put x = −5/2 into the left side.
We get 20, the same as the right side.
So both answers are right!
Let's look back.
First: divide by 2.
Then: complete the square.
Then: write it as a square.
Then: square root of both sides.
Then: get x alone.
Last: check.