Step by step
The question
Question: solve 4x² + 4x = 3 by completing the square.
1 Divide by 4
We want just x², not 4x².
So we divide every part by 4.
We do it on both sides, so they stay equal.
Some numbers become fractions.
That is fine.
Now we have x² + x = 3/4.
2 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.
3 Write it as a square
The full square is x + 1/2 on each side.
So the left side is (x + 1/2)².
Same bottom, so we add the tops: 3 plus 1 is 4.
That is 1.
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 = 1.
4 Square root of both sides
Something squared is 1.
So that something is the square root of 1, or minus it.
Both work, because minus times minus is plus.
The square root of 1 is 1, because 1 times 1 is 1.
5 Get x alone
To remove the + 1/2, we take away 1/2 on both sides.
We do it in both equations.
Same bottom, so we work with the tops.
1 minus 1/2 is 1/2.
Minus 1 minus 1/2 is −3/2.
So x = 1/2 or x = −3/2.
6 Check
Let's check: put x = 1/2 into the left side.
We get 3, the same as the right side.
It works!
Now put x = −3/2 into the left side.
We get 3, the same as the right side.
So both answers are right!
Let's look back.
First: divide by 4.
Then: complete the square.
Then: write it as a square.
Then: square root of both sides.
Then: get x alone.
Last: check.