Step by step
The question
Question: solve 2x² + 3x − 2 = 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 + 3x/2 − 1 = 0.
2 Move the −1
We want only the x parts on the left.
The − 1 is in the way.
So we add 1 on both sides.
− 1 and + 1 make zero.
Cross them out.
0 plus 1 is 1.
Now we have x² + 3x/2 = 1.
3 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.
4 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.
1 is 16/16.
Same bottom, so we add the tops: 16 plus 9 is 25.
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 = 25/16.
5 Square root of both sides
Something squared is 25/16.
So that something is the square root of 25/16, 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.
5 times 5 is 25.
So x + 3/4 is plus or minus that.
6 Get x alone
To remove the + 3/4, we take away 3/4 on both sides.
We do it in both equations.
Same bottom, so we work with the tops.
5/4 minus 3/4 is 1/2.
Minus 5/4 minus 3/4 is −2.
So x = 1/2 or x = −2.
7 Check
Let's check: put x = 1/2 into the left side.
We get 0, the same as the right side.
It works!
Now put x = −2 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 −1.
Then: complete the square.
Then: write it as a square.
Then: square root of both sides.
Then: get x alone.
Last: check.