Step by step
The question
Question: solve x² + 5x + 7 = 0 by completing the square.
1 Move the 7
We want only the x parts on the left.
The + 7 is in the way.
So we take away 7 on both sides.
+ 7 and − 7 make zero.
Cross them out.
0 minus 7 is −7.
Now we have x² + 5x = −7.
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.
−7 is −28/4.
Same bottom, so we add the tops: −28 plus 25 is −3.
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 = −3/4.
4 Square root of both sides
Something squared is −3/4.
But no real number squared is negative.
So there is no real solution.
Let's look back.
First: move the 7.
Then: complete the square.
Then: write it as a square.
Last: square root of both sides.