Step by step
The question
Question: solve x² + 3x − 5 = 0 by completing the square.
1 Move the −5
We want only the x parts on the left.
The − 5 is in the way.
So we add 5 on both sides.
− 5 and + 5 make zero.
Cross them out.
0 plus 5 is 5.
Now we have x² + 3x = 5.
2 Complete the square
We want the left side to be one bracket, squared.
A picture shows what is missing.
So we add 9/4 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 + 3/2 on each side.
So the left side is (x + 3/2)².
To add, both need the same bottom: 4.
5 is 20/4.
Same bottom, so we add the tops: 20 plus 9 is 29.
Careful!
Many people add 9/4 only on the left.
Then the two sides are not equal any more.
Now we have (x + 3/2)² = 29/4.
4 Square root of both sides
Something squared is 29/4.
So that something is the square root of 29/4, 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.
29 is not a square number, so we keep √29.
So x + 3/2 is plus or minus that.
5 Get x alone
To remove the + 3/2, we take away 3/2 on both sides.
We do it in both equations.
We write the plain number first.
So x = −3/2 + √29/2 or x = −3/2 − √29/2.
With a calculator, x is about 1.19 or about −4.19.
Let's look back.
First: move the −5.
Then: complete the square.
Then: write it as a square.
Then: square root of both sides.
Last: get x alone.