Completing the Square Solver
Enter any quadratic and see each step of completing the square — from rearranging the equation to writing the perfect-square form and solving for x.
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Completing the square — a full worked example
Below is one fully worked example plus a short primer so you can see exactly how our AI reasons through a problem.
Example Problem
DEMO- 1
Move the constant to the right side
Subtract 2 from both sides to isolate the variable terms.
- 2
Compute (b/2)² and add to both sides
b = 6, so (b/2)² = 3² = 9. Adding 9 to both sides keeps the equation balanced.
- 3
Factor the left side as a perfect-square trinomial
x² + 6x + 9 factors as (x + 3)². The right side is now a single number.
- 4
Take the square root of both sides
Remember both the positive and negative square root — that gives two solutions.
- 5
Solve for x
Subtract 3 from both sides. Check: for x = −3 + √7, (−3+√7)² + 6(−3+√7) + 2 = 9 − 6√7 + 7 − 18 + 6√7 + 2 = 0 ✓
Final Answer
How completing the square works — the technique behind the quadratic formula
Completing the square is an algebraic technique that rewrites a quadratic ax² + bx + c into the perfect-square form a(x + h)² + k. The process has three steps: move the constant term to the right side, add (b/2a)² to both sides, then factor the left side as a binomial square. From there, take square roots and solve for x. The method works for every quadratic — including ones whose roots are irrational or where factoring by inspection fails.
Beyond solving equations, completing the square converts standard form y = ax² + bx + c into vertex form y = a(x − h)² + k, which immediately reveals the parabola's turning point at (h, k). This matters in GCSE Higher, A-Level, SAT and JEE for questions on the minimum or maximum value of a quadratic. The quadratic formula itself is derived by completing the square on the general form — so mastering the method deepens your understanding of why the formula works, not just how to apply it.
Completing the square problems to practise (GCSE, SAT, Class 10)
Tap any problem to solve it with full step-by-step working.
- Solve with AI →1.Completing the square — monicClass 10 / GCSEEasy
- Solve with AI →2.Completing the square — non-monicClass 10 / GCSE HigherMedium
- Solve with AI →3.Vertex formGCSE / SATEasy
- Solve with AI →4.Optimisation — minimum valueA-Level / Class 11Medium
- Solve with AI →5.Word problem — projectileClass 11 / GCSE HigherHard
Frequently asked questions
What is completing the square and why is it used?+
Completing the square rewrites a quadratic ax² + bx + c as a(x + h)² + k. It is used for three things: solving quadratic equations where factoring fails, finding the vertex (turning point) of a parabola, and deriving the quadratic formula from first principles. It is a core technique in Class 10 algebra, GCSE Higher and SAT Maths.
How do I complete the square when the leading coefficient is not 1?+
Divide the entire equation by a first to make the coefficient of x² equal to 1, then follow the standard steps: half the coefficient of x, square it, add and subtract on the left, factor. For example, 2x² − 12x + 7 becomes x² − 6x + 3.5 = 0, and then (x − 3)² = 5.5, giving x = 3 ± √5.5.
How does completing the square relate to the quadratic formula?+
The quadratic formula is derived directly by completing the square on ax² + bx + c = 0. Dividing by a and completing the square gives (x + b/2a)² = (b² − 4ac) / 4a², which simplifies to x = (−b ± √(b² − 4ac)) / 2a. Understanding this derivation shows why the discriminant b² − 4ac controls the nature of the roots.
What is vertex form and how does completing the square give it?+
Vertex form is y = a(x − h)² + k, where (h, k) is the turning point of the parabola. You get it by completing the square on the standard form y = ax² + bx + c. The vertex form immediately tells you the minimum or maximum value of the quadratic (k when a > 0 gives a minimum, a < 0 gives a maximum) without needing calculus.
When should I use completing the square instead of factoring or the quadratic formula?+
Use completing the square when the coefficient of x² is 1 and b is even, since the arithmetic stays clean. It is also the method to choose when a question explicitly asks for vertex form or the minimum/maximum value of a quadratic expression. The quadratic formula is faster for generic numerical answers; completing the square is better for understanding the structure.
Is completing the square on the GCSE, SAT and A-Level syllabus?+
Yes, on all three. GCSE Higher (AQA/Edexcel/OCR) requires students to complete the square to solve quadratics and find the vertex. SAT Maths includes completing the square for vertex form questions in the algebra section. A-Level and Class 11/12 extend it to surds, complex roots and quadratic inequalities.
Is there a free completing the square calculator with full steps?+
Yes — this solver shows every step: rearranging, adding the square term, factoring the trinomial, taking square roots and solving. The free plan gives 5 daily solves with no credit card needed. Steps are never paywalled.
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