Worked examples
The question
Question: solve x² + 3x = 10 by completing the square.
x2 + 3x = 10
1 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.
x2 + 3x + 94 = 10 + 94
2 Write it as a square
The full square is x + 3/2 on each side.
So the left side is (x + 3/2)².
x2 + 3x + 94 = (x + 32)2
To add, both need the same bottom: 4.
10 is 40/4.
Same bottom, so we add the tops: 40 plus 9 is 49.
10 = 404
404 + 94 = 494
Careful!
Many people add 9/4 only on the left.
Then the two sides are not equal any more.
Careful: add 94 on BOTH sides
Now we have (x + 3/2)² = 49/4.
(x + 32)2 = 494
3 Square root of both sides
Something squared is 49/4.
So that something is the square root of 49/4, or minus it.
Both work, because minus times minus is plus.
( )² = k → ( ) = √k or −√k
The square root of a fraction is the root of the top over the root of the bottom.
2 times 2 is 4.
7 times 7 is 49.
√494 = √49√4 = 72
So x + 3/2 is plus or minus that.
x + 32 = 72 or x + 32 = −72
4 Get x alone
To remove the + 3/2, we take away 3/2 on both sides.
We do it in both equations.
x = 72 − 32 or x = −72 − 32
Same bottom, so we work with the tops.
7/2 minus 3/2 is 2.
Minus 7/2 minus 3/2 is −5.
72 − 32 = 42 = 2
−72 − 32 = −102 = −5
So x = 2 or x = −5.
x = 2 or x = −5
5 Check
Let's check: put x = 2 into the left side.
x = 2: 22 + 3(2) = 4 + 6 = 10
We get 10, the same as the right side.
It works!
Now put x = −5 into the left side.
x = −5: (−5)2 + 3(−5) = 25 − 15 = 10
We get 10, the same as the right side.
So both answers are right!
Let's look back.
First: complete the square.
Then: write it as a square.
Then: square root of both sides.
Then: get x alone.
Last: check.
▶ Watch it solved
The question
Question: solve x² + 3x − 4 = 0 by completing the square.
x2 + 3x − 4 = 0
1 Move the −4
We want only the x parts on the left.
The − 4 is in the way.
So we add 4 on both sides.
x2 + 3x − 4 + 4 = 0 + 4
− 4 and + 4 make zero.
Cross them out.
0 plus 4 is 4.
0 + 4 = 4
Now we have x² + 3x = 4.
x2 + 3x = 4
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.
x2 + 3x + 94 = 4 + 94
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.
x2 + 3x + 94 = (x + 32)2
To add, both need the same bottom: 4.
4 is 16/4.
Same bottom, so we add the tops: 16 plus 9 is 25.
4 = 164
164 + 94 = 254
Careful!
Many people add 9/4 only on the left.
Then the two sides are not equal any more.
Careful: add 94 on BOTH sides
Now we have (x + 3/2)² = 25/4.
(x + 32)2 = 254
4 Square root of both sides
Something squared is 25/4.
So that something is the square root of 25/4, or minus it.
Both work, because minus times minus is plus.
( )² = k → ( ) = √k or −√k
The square root of a fraction is the root of the top over the root of the bottom.
2 times 2 is 4.
5 times 5 is 25.
√254 = √25√4 = 52
So x + 3/2 is plus or minus that.
x + 32 = 52 or x + 32 = −52
5 Get x alone
To remove the + 3/2, we take away 3/2 on both sides.
We do it in both equations.
x = 52 − 32 or x = −52 − 32
Same bottom, so we work with the tops.
5/2 minus 3/2 is 1.
Minus 5/2 minus 3/2 is −4.
52 − 32 = 22 = 1
−52 − 32 = −82 = −4
So x = 1 or x = −4.
x = 1 or x = −4
6 Check
Let's check: put x = 1 into the left side.
x = 1: 12 + 3(1) − 4 = 1 + 3 − 4 = 0
We get 0, the same as the right side.
It works!
Now put x = −4 into the left side.
x = −4: (−4)2 + 3(−4) − 4 = 16 − 12 − 4 = 0
We get 0, the same as the right side.
So both answers are right!
Let's look back.
First: move the −4.
Then: complete the square.
Then: write it as a square.
Then: square root of both sides.
Then: get x alone.
Last: check.
▶ Watch it solved
The question
Question: solve 2x² + 3x − 2 = 0 by completing the square.
2x2 + 3x − 2 = 0
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.
2x² ÷ 2 = x²
Some numbers become fractions.
That is fine.
Now we have x² + 3x/2 − 1 = 0.
x2 + 3x2 − 1 = 0
2 Move the −1
We want only the x parts on the left.
The − 1 is in the way.
So we add 1 on both sides.
x2 + 3x2 − 1 + 1 = 0 + 1
− 1 and + 1 make zero.
Cross them out.
0 plus 1 is 1.
0 + 1 = 1
Now we have x² + 3x/2 = 1.
x2 + 3x2 = 1
3 Complete the square
We want the left side to be one bracket, squared.
A picture shows what is missing.
So we add 9/16 to fill the corner.
We do it on both sides, so they stay equal.
x2 + 3x2 + 916 = 1 + 916
4 Write it as a square
The full square is x + 3/4 on each side.
So the left side is (x + 3/4)².
x2 + 3x2 + 916 = (x + 34)2
To add, both need the same bottom: 16.
1 is 16/16.
Same bottom, so we add the tops: 16 plus 9 is 25.
1 = 1616
1616 + 916 = 2516
Careful!
Many people add 9/16 only on the left.
Then the two sides are not equal any more.
Careful: add 916 on BOTH sides
Now we have (x + 3/4)² = 25/16.
(x + 34)2 = 2516
5 Square root of both sides
Something squared is 25/16.
So that something is the square root of 25/16, or minus it.
Both work, because minus times minus is plus.
( )² = k → ( ) = √k or −√k
The square root of a fraction is the root of the top over the root of the bottom.
4 times 4 is 16.
5 times 5 is 25.
√2516 = √25√16 = 54
So x + 3/4 is plus or minus that.
x + 34 = 54 or x + 34 = −54
6 Get x alone
To remove the + 3/4, we take away 3/4 on both sides.
We do it in both equations.
x = 54 − 34 or x = −54 − 34
Same bottom, so we work with the tops.
5/4 minus 3/4 is 1/2.
Minus 5/4 minus 3/4 is −2.
54 − 34 = 24 = 12
−54 − 34 = −84 = −2
So x = 1/2 or x = −2.
x = 12 or x = −2
7 Check
Let's check: put x = 1/2 into the left side.
x = 12: 2(12)2 + 3(12) − 2 = 12 + 32 − 2 = 0
We get 0, the same as the right side.
It works!
Now put x = −2 into the left side.
x = −2: 2(−2)2 + 3(−2) − 2 = 8 − 6 − 2 = 0
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 −1.
Then: complete the square.
Then: write it as a square.
Then: square root of both sides.
Then: get x alone.
Last: check.
▶ Watch it solved
