Completing the Square Calculator

The Completing the Square Calculator converts any equation of the form ax² + bx + c = 0 into vertex form a(x − h)² + k, and solves for x by completing the square rather than jumping straight to the quadratic formula. Enter the three coefficients — a, b, and c — and it returns the vertex form, the vertex point (h, k), the roots, and a full step-by-step derivation showing exactly how each is reached.

It's built for algebra students learning the completing-the-square method itself (not just the answer), teachers preparing worked examples, and anyone who needs a parabola's vertex form without doing the algebra by hand.

How to use it

1. Enter the coefficient a (the number multiplying x²). It must not be zero.
2. Enter the coefficient b (the number multiplying x).
3. Enter the coefficient c (the constant term).
4. Read Vertex Form and Vertex (h, k) for the converted equation, Roots for the solution, and Step-by-Step Solution for the full derivation.

Results update automatically as you type. For example, entering a = 1, b = 6, c = 5 (for x² + 6x + 5 = 0) returns vertex form y = (x + 3)² - 4, vertex (-3, -4), and roots x₁ = -1, x₂ = -5.

What is completing the square?

Completing the square rewrites ax² + bx + c by factoring a out of the x² and x terms, then adding and subtracting the square of half the x-coefficient so the expression inside the parentheses becomes a perfect square trinomial:

ax² + bx + c
= a(x² + (b/a)x) + c
= a(x² + (b/a)x + (b/2a)²) − a(b/2a)² + c
= a(x + b/2a)² + (c − b²/4a)
= a(x − h)² + k,   where h = −b/(2a),  k = c − b²/(4a)

That final line is vertex form — for the parabola y = ax² + bx + c, the point (h, k) is its vertex (the minimum if a > 0, the maximum if a < 0).

How does it solve for x?

Setting the vertex form to zero and isolating the squared term:

a(x − h)² + k = 0  →  (x − h)² = −k/a

• If −k/a > 0, there are two distinct real roots: x = h ± √(−k/a).
• If −k/a = 0, there's one repeated real root: x = h.
• If −k/a < 0, there are two complex conjugate roots: x = h ± i√(k/a).

These are exactly the same roots the quadratic formula produces for the same a, b, c — completing the square is a different route to the same answer, and is in fact how the quadratic formula itself is derived. For solving via the discriminant directly, see the Quadratic Formula Solver.

Why can't a be zero?

If a = 0, the x² term disappears and the equation reduces to bx + c = 0 — a linear equation with no vertex to complete the square toward. The calculator flags a = 0 as invalid input rather than silently switching to linear-equation logic.

How are complex roots formatted?

Complex roots are shown in standard a + bi notation, for example -0.5 + 0.866025i and its conjugate -0.5 - 0.866025i — the real part is the vertex's h, and the imaginary part comes from √(k/a).

Privacy

This calculator runs entirely in your browser. Your coefficients are never uploaded, logged, or stored — the computation happens locally on your device.

Related tools

For solving the same equation via the discriminant instead, see the Quadratic Formula Solver.