Completing the Square Calculator
Enter the coefficients of your quadratic equation ax² + bx + c to get the completed square form, vertex, and step-by-step working.
Updated: February 2026 | For UK GCSE & A-Level Maths | Author: Mustafa Bilgic (MB)
Completing the Square Calculator
Enter the values of a, b, and c for your quadratic expression ax² + bx + c. The calculator will complete the square and show all working.
What Is Completing the Square?
Completing the square is an algebraic technique used to rewrite a quadratic expression ax² + bx + c into the equivalent form a(x + p)² + q. This is called the completed square form or vertex form of the quadratic. It is one of the most important techniques in GCSE and A-Level Mathematics, underpinning the derivation of the quadratic formula and the analysis of parabolas.
The completed square form is particularly powerful because:
- It immediately reveals the vertex (turning point) of the parabola at (−p, q)
- It shows the axis of symmetry at x = −p
- It allows equations to be solved without the quadratic formula
- It determines the nature of roots via the sign of q/a
- It is used in integration and in solving problems involving maximum and minimum values
The Method: Step by Step
For any quadratic ax² + bx + c, here is the complete method:
Step 1: Factor out 'a' from the first two terms (if a ≠ 1)
Write: a(x² + (b/a)x) + c. This isolates the squared and linear terms inside a bracket with a coefficient of 1 on x².
Step 2: Find the completing term
Inside the bracket, take half the coefficient of x: p = b/(2a). The term that completes the square is p².
Step 3: Add and subtract p² inside the bracket
Write: a((x + p)² − p²) + c. The expression (x + p)² − p² is identical to x² + (b/a)x, so nothing has changed in value.
Step 4: Expand the bracket around a and simplify
Distribute the 'a': a(x + p)² − ap² + c. Combine the constants: q = c − ap² = c − b²/(4a).
Step 5: Write the final completed square form
Result: a(x + p)² + q, where p = b/(2a) and q = c − b²/(4a). The vertex of the parabola is at (−p, q).
Finding Roots from the Completed Square Form
Once you have a(x + p)² + q = 0, solving for x is straightforward:
(x + p)² = −q/a
x + p = ±√(−q/a)
x = −p ± √(−q/a)
The discriminant (determining number of real roots) is equivalent to b² − 4ac. If −q/a < 0, no real roots exist. If −q/a = 0, there is one repeated root. If −q/a > 0, there are two distinct real roots.
Worked Example 1: x² + 6x + 5
Here a = 1, b = 6, c = 5:
Step 1: a = 1, so no factoring needed
Step 2: p = 6/(2×1) = 3
Step 3: (x + 3)² − 9
Step 4: (x + 3)² − 9 + 5
Completed: (x + 3)² − 4
Vertex: (−3, −4)
Roots: (x + 3)² = 4 → x + 3 = ±2 → x = −1 or x = −5
Worked Example 2: 2x² − 8x + 3 (a ≠ 1)
Here a = 2, b = −8, c = 3:
Step 1: Factor out 2: 2(x² − 4x) + 3
Step 2: p = −4/(2×1) = −2 (inside bracket, coeff of x is −4)
Step 3: 2((x − 2)² − 4) + 3
Step 4: 2(x − 2)² − 8 + 3
Completed: 2(x − 2)² − 5
Vertex: (2, −5)
Roots: 2(x−2)² = 5 → (x−2)² = 2.5 → x = 2 ± √2.5
Applications of Completing the Square
Beyond solving equations, completing the square has wide-ranging applications in UK Mathematics syllabuses:
- Curve Sketching (A-Level): Vertex form directly gives the turning point and axis of symmetry, making it easy to sketch parabolas accurately.
- Deriving the Quadratic Formula: The quadratic formula x = (−b ± √(b²−4ac)) / 2a is derived by completing the square on the general form ax² + bx + c = 0.
- Optimisation Problems: Finding maximum profit, minimum cost, or maximum area all require identifying the vertex of a quadratic — achieved via completing the square.
- Circle Equations (A-Level): The equation of a circle x² + y² + 2gx + 2fy + c = 0 can be converted to centre-radius form by completing the square on both x and y terms.
- Integration by Substitution (A-Level): Completing the square simplifies integrals of the form ∫1/(ax²+bx+c) dx into standard arctangent or logarithm forms.
Common Mistakes to Avoid
- Forgetting to multiply p² by a: When a ≠ 1, the constant subtracted when distributing 'a' back is ap², not just p².
- Sign errors with negative b: If b is negative, p = b/(2a) is negative, meaning the bracket becomes (x + p)² where p is negative — equivalent to (x − |p|)².
- Confusing vertex coordinates: The vertex is at (−p, q) — note the negative sign. If the form is (x + 3)², the x-coordinate of the vertex is −3, not +3.
- Ignoring 'a' when finding the vertex y-coordinate: The y-coordinate of the vertex (the value of q) is c − b²/(4a), not simply c − p².
Frequently Asked Questions: Completing the Square
What does completing the square mean in GCSE Maths?
Completing the square is a method of rewriting a quadratic expression ax² + bx + c into the form a(x + p)² + q. This reveals the vertex of the parabola at (−p, q), allows the equation to be solved without the quadratic formula, and shows the axis of symmetry at x = −p. It is a core GCSE and A-Level technique.
How do you complete the square step by step?
For ax² + bx + c: (1) Factor 'a' from the first two terms. (2) Find p = b/(2a). (3) Write a(x + p)² − ap². (4) Add c to get a(x + p)² + (c − ap²). The completed form is a(x + p)² + q where q = c − b²/(4a). The vertex is at (−p, q).
How do you find the roots using completing the square?
From a(x + p)² + q = 0: rearrange to (x + p)² = −q/a. Take square roots: x + p = ±√(−q/a). So x = −p ± √(−q/a). If −q/a is negative, there are no real roots. If zero, one repeated root. If positive, two distinct roots.
What is the vertex of a parabola?
The vertex is the turning point — minimum if a > 0 (parabola opens up) or maximum if a < 0 (opens down). In completed square form a(x + p)² + q, the vertex is at (−p, q). For 2(x + 3)² − 5, the vertex is at (−3, −5).
When should I use completing the square vs the quadratic formula?
Use completing the square when you need the vertex, axis of symmetry, or the question explicitly asks for vertex form. The quadratic formula is faster for numerical root-finding with complex coefficients. Both methods are equivalent; the quadratic formula is derived by completing the square on the general form.
Does completing the square work when a ≠ 1?
Yes. Factor 'a' out of the first two terms first. For 2x² + 8x + 3: write 2(x² + 4x) + 3. Complete the square on x² + 4x = (x+2)² − 4. Then: 2((x+2)² − 4) + 3 = 2(x+2)² − 8 + 3 = 2(x+2)² − 5. Vertex: (−2, −5).
What is the discriminant and what does it tell us?
The discriminant b² − 4ac determines the nature of roots. If positive: two distinct real roots. If zero: one repeated root. If negative: no real roots (complex). In completed square form, this corresponds to the sign of −q/a determining whether real roots exist.
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