To complete the square, move the constant to one side of the equation, take half the coefficient of x, square it, and add that value to both sides. This turns the left side into a perfect square binomial, which you can solve by taking the square root of both sides. It works on every quadratic equation, even ones that cannot be factored.
What Does It Mean to Complete the Square?
Completing the square is a method for rewriting a quadratic expression so it contains a perfect square binomial instead of two separate x-terms. A quadratic like x² + 6x is not a perfect square on its own, but adding 9 turns it into x² + 6x + 9, which factors neatly as (x + 3)².
Beyond solving for x, this same technique shows up when graphing a parabola or finding its highest or lowest point, since vertex form puts that information directly in the equation. It also underlies the derivation of the quadratic formula, so a solid grasp of completing the square makes most other quadratic topics easier to follow.
The Completing the Square Formula
The completing the square formula starts from the standard quadratic form:
ax² + bx + c = 0
To complete the square, rearrange it into:
a(x + b/2a)² = (b² − 4ac) / 4a
The key number to remember is (b/2)², or half the coefficient of x, squared. This is the value that turns x² + bx into a perfect square trinomial. In fact, the Common Core State Standards for high school algebra specifically require students to use this method to transform any quadratic equation into the form (x − p)² = q, and then derive the quadratic formula from that same transformation.
A quick way to check your work is to expand the result back out. If a(x + b/2a)² simplifies back to the original ax² + bx + c, you completed the square correctly. This habit catches the most common error in this method: adding the constant to only one side instead of both.
How to Complete the Square Step by Step
Here is how to complete the square step by step, using x² + 8x + 7 = 0 as a working example.
- Move the constant to the other side. Subtract 7 from both sides: x² + 8x = −7.
- Take half the coefficient of x, then square it. Half of 8 is 4, and 4² = 16.
- Add that value to both sides. x² + 8x + 16 = −7 + 16, which simplifies to x² + 8x + 16 = 9.
- Factor the left side as a perfect square. x² + 8x + 16 factors into (x + 4)².
- Take the square root of both sides. (x + 4)² = 9 becomes x + 4 = ±3.
- Solve for x. x = −4 + 3 = −1, or x = −4 − 3 = −7.
When the coefficient of x² is not 1
If the equation looks like 2x² + 12x + 10 = 0, divide every term by the leading coefficient first. Dividing by 2 gives x² + 6x + 5 = 0, and from there, the same six steps apply exactly as before. This division step is the one most students forget, so check for a leading coefficient before doing anything else. Seeing this complete-the-square method applied to a few different coefficients in a step-by-step video explanation makes the pattern automatic faster than the rule alone.
Complete the Square Method: Worked Examples
Example 1. Solving a quadratic equation
Solve x² − 4x − 5 = 0 using the complete-the-square method.
Move the constant: x² − 4x = 5. Half of −4 is −2, and (−2)² = 4. Add 4 to both sides: x² − 4x + 4 = 9. Factor: (x − 2)² = 9. Take the square root: x − 2 = ±3. Solve: x = 5 or x = −1.
Example 2. A coefficient other than 1
Solve 3x² + 12x − 15 = 0.
Divide everything by 3: x² + 4x − 5 = 0. Move the constant: x² + 4x = 5. Half of 4 is 2, and 2² = 4. Add 4 to both sides: x² + 4x + 4 = 9. Factor: (x + 2)² = 9. Take the square root: x + 2 = ±3. Solve: x = 1 or x = −5.
Example 3. Converting to vertex form
Rewrite f(x) = x² − 6x + 2 in vertex form.
Half of −6 is −3, and (−3)² = 9. Rewrite as x² − 6x + 9 − 9 + 2, which groups into (x − 3)² − 7. The vertex form is f(x) = (x − 3)² − 7, showing the parabola’s vertex sits at (3, −7). Watching this rearrangement happen one line at a time in a narrated video walkthrough makes the regrouping step click faster than reading it cold.
Example 4. A word problem using vertex form
A ball is thrown so its height in meters is modeled by h(t) = −5t² + 20t, where t is time in seconds. Find the maximum height.
Factor out −5: h(t) = −5(t² − 4t). Half of −4 is −2, and (−2)² = 4. Add and subtract inside the parentheses: h(t) = −5(t − 2)² + 20. The vertex is at (2, 20), so the ball reaches a maximum height of 20 meters at t = 2 seconds. This is where completing the square beats factoring, since the vertex gives the maximum directly.
Completing the Square Practice Problems
Try these completing-the-square practice problems, then check your work against the solutions below.
Problem 1. Solve x² + 10x + 21 = 0. Solution: x² + 10x = −21. Half of 10 is 5, and 5² = 25. x² + 10x + 25 = 4. Factor: (x + 5)² = 4. So x + 5 = ±2, giving x = −3 or x = −7.
Problem 2. Solve 2x² − 8x − 24 = 0. Solution: Divide by 2: x² − 4x − 12 = 0. Move the constant: x² − 4x = 12. Half of −4 is −2, and (−2)² = 4. x² − 4x + 4 = 16. Factor: (x − 2)² = 16. So x − 2 = ±4, giving x = 6 or x = −2.
Problem 3. Convert f(x) = x² + 2x − 8 to vertex form. Solution: Half of 2 is 1, and 1² = 1. Rewrite as x² + 2x + 1 − 1 − 8, which groups into (x + 1)² − 9.
Checklist before you submit an answer
- Did you divide by the leading coefficient if it wasn’t 1?
- Did you add the same value to both sides, not just one?
- Did you factor the left side correctly as a perfect square binomial?
- Did you include both the positive and negative square root when solving for x?
If any of these steps feel shaky, working through a worked video example of a similar problem before your next practice set can catch the gap early.
How Think10x.ai Helps You Master Completing the Square
Completing the square trips students up not because the concept is hard, but because it has several small moving parts such as dividing by the leading coefficient, halving and squaring the right term, and keeping both sides balanced.
Think10x.ai turns a completing-the-square question, typed, spoken, or photographed from a worksheet, into a narrated video that walks through each step individually, and students can pause and ask why a step happens instead of just copying it down.
Teachers preparing practice sets can generate full walkthroughs using the platform’s AI math solver, and students working through homework independently get the same explanation depth through the Think10x.ai homepage, without relying on a static answer key alone.
Frequently Asked Questions
Yes. Unlike factoring, which only works cleanly on certain quadratics, completing the square works on every quadratic equation, including ones with irrational or complex solutions.
They are closely related. The quadratic formula is derived by completing the square on ax² + bx + c = 0, so learning one deepens your understanding of the other.
Adding a number only to one side would change the equation’s value. Adding it to both sides keeps the equation balanced while still turning the left side into a perfect square.
Yes. The equation then has complex solutions instead of real ones, written using i, the imaginary unit, since you cannot take the square root of a negative real number.
No. It is also used to convert a quadratic function into vertex form, which makes it easy to read off a parabola’s maximum or minimum point directly, without graphing first.