Step by step
The question
Question: solve 2x² − 5x + 3 = 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 − 5x/2 + 3/2 = 0.
2 Move the plain number
We want only the x parts on the left.
The + 3/2 is in the way.
So we take away 3/2 on both sides.
+ 3/2 and − 3/2 make zero.
Cross them out.
0 minus 3/2 is −3/2.
Now we have x² − 5x/2 = −3/2.
3 Complete the square
We want the left side to be one bracket, squared.
A picture shows what is missing.
So we add 25/16 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 − 5/4 on each side.
So the left side is (x − 5/4)².
To add, both need the same bottom: 16.
−3/2 is −24/16.
Same bottom, so we add the tops: −24 plus 25 is 1.
Careful!
Many people add 25/16 only on the left.
Then the two sides are not equal any more.
Now we have (x − 5/4)² = 1/16.
5 Square root of both sides
Something squared is 1/16.
So that something is the square root of 1/16, 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.
1 times 1 is 1.
So x − 5/4 is plus or minus that.
6 Get x alone
To remove the − 5/4, we add 5/4 on both sides.
We do it in both equations.
Same bottom, so we work with the tops.
1/4 plus 5/4 is 3/2.
Minus 1/4 plus 5/4 is 1.
So x = 3/2 or x = 1.
7 Check
Let's check: put x = 3/2 into the left side.
We get 0, the same as the right side.
It works!
Now put x = 1 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 plain number.
Then: complete the square.
Then: write it as a square.
Then: square root of both sides.
Then: get x alone.
Last: check.