Step by step
The question
Question: solve 3x² + 2x = 1 by completing the square.
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.
Some numbers become fractions.
That is fine.
Now we have x² + 2x/3 = 1/3.
2 Complete the square
We want the left side to be one bracket, squared.
A picture shows what is missing.
So we add 1/9 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 + 1/3 on each side.
So the left side is (x + 1/3)².
To add, both need the same bottom: 9.
1/3 is 3/9.
Same bottom, so we add the tops: 3 plus 1 is 4.
Careful!
Many people add 1/9 only on the left.
Then the two sides are not equal any more.
Now we have (x + 1/3)² = 4/9.
4 Square root of both sides
Something squared is 4/9.
So that something is the square root of 4/9, 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.
3 times 3 is 9.
2 times 2 is 4.
So x + 1/3 is plus or minus that.
5 Get x alone
To remove the + 1/3, we take away 1/3 on both sides.
We do it in both equations.
Same bottom, so we work with the tops.
2/3 minus 1/3 is 1/3.
Minus 2/3 minus 1/3 is −1.
So x = 1/3 or x = −1.
6 Check
Let's check: put x = 1/3 into the left side.
We get 1, the same as the right side.
It works!
Now put x = −1 into the left side.
We get 1, 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.