Step by step
The question
Question: solve x² − 5x + 2 = 0 by completing the square.
1 Move the 2
We want only the x parts on the left.
The + 2 is in the way.
So we take away 2 on both sides.
+ 2 and − 2 make zero.
Cross them out.
0 minus 2 is −2.
Now we have x² − 5x = −2.
2 Complete the square
We want the left side to be one bracket, squared.
A picture shows what is missing.
So we add 25/4 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 − 5/2 on each side.
So the left side is (x − 5/2)².
To add, both need the same bottom: 4.
−2 is −8/4.
Same bottom, so we add the tops: −8 plus 25 is 17.
Careful!
Many people add 25/4 only on the left.
Then the two sides are not equal any more.
Now we have (x − 5/2)² = 17/4.
4 Square root of both sides
Something squared is 17/4.
So that something is the square root of 17/4, 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.
2 times 2 is 4.
17 is not a square number, so we keep √17.
So x − 5/2 is plus or minus that.
5 Get x alone
To remove the − 5/2, we add 5/2 on both sides.
We do it in both equations.
We write the plain number first.
So x = 5/2 + √17/2 or x = 5/2 − √17/2.
With a calculator, x is about 4.56 or about 0.44.
Let's look back.
First: move the 2.
Then: complete the square.
Then: write it as a square.
Then: square root of both sides.
Last: get x alone.