Completing the square rewrites x² + bx + c as (x + p)² + q. Halve b to get p, then take away p² so the value stays the same. It solves quadratics, finds turning points and tidies circle equations.
Add the missing corner, then take it away.
x² + 6x = (x + 3)² − 9
Hello, curious mind!
What is completing the square?
Completing the square means writing x² + bx + c as a perfect square plus a number: (x + b/2)² − (b/2)² + c.
A perfect square is a bracket times itself, such as (x + 3)². In the picture, x² and two strips of 3x nearly make a square of side x + 3. Only the 3 × 3 corner is missing. Add it to finish the square, then take 9 away so nothing changes. For any b: halve it, square the half, add it and take it away.
Remember this
x² + bx = (x + b/2)² − (b/2)²
How to complete the square: 3 worked examples
The steps: make the x² number 1, halve the x number (keep its sign), write the bracket, take away the half squared, then tidy up.
Example 1 (a = 1): solve x² − 14x + 37 = 0
Half of −14 is −7. The bracket is (x − 7)², which is x² − 14x + 49.
Take away 49: x² − 14x + 37 = (x − 7)² − 49 + 37 = (x − 7)² − 12.
So (x − 7)² = 12, and x − 7 = ±√12.
x = 7 ± 2√3, which is about 10.46 or 3.54.
Example 2 (a ≠1): solve 2x² − 8x − 5 = 0
Divide every term by 2 first: x² − 4x − 2.5 = 0.
Half of −4 is −2. So (x − 2)² − 4 − 2.5 = 0, which gives (x − 2)² = 6.5.
x = 2 ± √6.5, which is about 4.55 or −0.55.
For the form only, take the 2 outside: 2[(x − 2)² − 4] − 5 = 2(x − 2)² − 13. The 4 is multiplied by 2 too.
Example 3 (turning point): y = x² − 5x + 4
Half of −5 is −2.5, and (−2.5)² = 6.25. So y = (x − 2.5)² − 2.25.
A square is never below 0, so y is smallest when x = 2.5. The turning point is (2.5, −2.25).
Check: the roots are 1 and 4, and 2.5 is halfway between them.
The 10-second check: expand it back
Expand your answer. It must give what you started with: (x − 7)² − 12 = x² − 14x + 49 − 12 = x² − 14x + 37.
Faster: put x = 0 into both forms. (0 − 7)² − 12 = 37, the last number of x² − 14x + 37. This catches a lost “take away” step, but not a wrong sign in the bracket.
Where the quadratic formula comes from
Complete the square on ax² + bx + c = 0 with letters.
Divide by a. Half of b/a is b/2a, so (x + b/2a)² − b²/4a² + c/a = 0.
Move the numbers across: (x + b/2a)² = (b² − 4ac) ÷ 4a².
Square root both sides: x = (−b ± √(b² − 4ac)) ÷ 2a.
So b² − 4ac decides how many roots there are. See the quadratic formula and quadratic equations pages to pick the fastest method.
Common completing the square mistakes
- Not taking away the extra number. x² + 8x + 3 is not (x + 4)² + 3. The bracket adds 16, so it is (x + 4)² − 13.
- Not dealing with a first. 3x² − 12x + 2 is not (x − 6)² − 34. Take 3 out: 3(x − 2)² − 10.
- Losing the sign when you halve. In x² − 10x + 4, half of −10 is −5. So it is (x − 5)² − 21, not (x + 5)² − 21.
- Taking only the positive root. (x − 1)² = 25 gives x − 1 = ±5, so x = 6 or x = −4.
- Doing steps in your head. If the steps are not written, nobody can find the slip.
Check this with the Free BMI Maths Test
- Forgets to take away (b/2)² again: CNC, concept not clear.
- Does not divide by a, or take it out, first: CNC, concept not clear.
- Halves a negative b and gets the wrong sign: CMT, calculation mistake.
- Takes only the positive square root: CNC, concept not clear.
- Drops steps, so the answer cannot be checked: BPT, bad presentation.
The Free BMI Maths Test finds which of the 6 mistake types your child makes, and their real maths level. About the Free BMI Maths Test
Test prep corner
Completing the square in SAT and JEE Main maths
College Board lists “Equivalent expressions” and “Circles” in SAT maths. The SAT reference sheet gives a circle’s area and circumference, but not its equation. NTA’s JEE Main maths topic list names quadratic equations and the general form of a circle’s equation. These are SAT-style and JEE-style practice questions, not real past questions.
SAT-style: A circle in the xy-plane has the equation x² + y² + 4x − 12y = 9. Its area is kπ. What is k?
Complete the square twice: (x + 2)² + (y − 6)² = 9 + 4 + 36 = 49. The radius is 7, so the area is 49π and k = 49. Trap: answering 9, the number on the right before you complete the square.
JEE-style: For real x, what is the greatest value of 1 ÷ (x² − 6x + 14)?
x² − 6x + 14 = (x − 3)² + 5, so the bottom is never less than 5. The smallest bottom gives the biggest fraction: 1/5, at x = 3. Trap: looking for the greatest value of the bottom.
Try harder ones in our hard SAT-style maths questions for Module 2.
Facts checked on 11 November 2026 on satsuite.collegeboard.org and jeemain.nta.nic.in.
Quick check: 5 completing the square questions
1. What number do you add to x² + 18x to make a perfect square?
Not quite. 9 is half of 18. Now square it.
Yes! x² + 18x + 81 = (x + 9)².
Not quite. Halve 18, then square.
2. Which is x² − 2x + 7 in completed square form?
Yes! (x − 1)² adds 1, so 7 − 1 = 6.
Not quite. Take the 1 away, do not add it.
Not quite. Half of −2 is −1, so the bracket is (x − 1).
3. What is the turning point of y = (x + 4)² − 3?
Not quite. The bracket is 0 when x = −4.
Yes! x = −4 makes the bracket 0, so y = −3.
Not quite. The x-value comes from the bracket.
4. Solve (x + 2)² = 9.
Not quite. x + 2 can also be −3.
Not quite. Take the square root of 9 first: 3, not 9.
Yes! x + 2 = 3 or x + 2 = −3.
5. 2x² + 8x + 1 = 2(x + 2)² + q. What is q?
Not quite. Multiply the 4 by 2 as well.
Yes! 2(x + 2)² = 2x² + 8x + 8, so q = −7.
Not quite. Square the 2 in the bracket, not the 4.
Got 2 or fewer right? Book the Free BMI Maths Test to find the cause.
Questions people ask about completing the square
What is completing the square?
It writes x² + bx + c as (x + b/2)² plus a number. Halve b, square it, add it, then take it away.
When should I use it instead of the formula?
Use it for a turning point, a smallest or greatest value, or a circle’s centre and radius. For roots only, the formula is often faster.
How do I complete the square when a is not 1?
To solve, divide every term by a first. To rewrite, take a outside a bracket, and multiply the number you take away by a.
How does it find the turning point?
In y = (x + p)² + q, the turning point is (−p, q). In y = −(x − 3)² + 10, the minus sign makes (3, 10) a highest point.
Is completing the square on the SAT?
Yes, mainly in circle questions. They often give x² + y² + Dx + Ey = F, and you complete the square to find the centre and radius.
Does the extra number keep going missing?
The Free BMI Maths Test finds your child’s real maths level and the reason for each mistake.
