Completing the Square Calculator
Convert ax² + bx + c to vertex form a(x-h)² + k by completing the square.
Convert ax² + bx + c to vertex form a(x − h)² + k
What Is the Completing the Square Calculator?
The Completing the Square Calculator rewrites any quadratic expression ax² + bx + c in vertex form a(x − h)² + k. It reveals the vertex coordinates, axis of symmetry, and roots of the parabola — and provides a step-by-step explanation of the completing-the-square process.
Formula
How to Use
Enter the coefficients a, b, and c of the quadratic expression ax² + bx + c. The calculator completes the square algebraically, showing each step: factoring out a, halving b/a, adding and subtracting the square, and expressing in vertex form.
Example Calculation
Complete the square for x² + 6x + 5: h = −6/2 = −3; k = 5 − 9 = −4. Vertex form: (x+3)² − 4. Vertex: (−3, −4). Roots: x+3 = ±2 → x = −1 or x = −5. Check: x²+6x+5 = (x+1)(x+5) ✓
Understanding Completing the Square
Completing the square is an algebraic technique that transforms a quadratic expression from standard form (ax² + bx + c) to vertex form (a(x−h)² + k). This rewriting reveals the parabola's vertex at (h, k), its axis of symmetry at x = h, and makes solving for roots straightforward — just isolate the squared term.
The process involves halving the coefficient of x, squaring it, and adding and subtracting this value to maintain equality. When a ≠ 1, the leading coefficient must first be factored out. The result is a perfect square trinomial plus a constant adjustment that gives the vertex height k.
Beyond parabola analysis, completing the square is used to derive the quadratic formula, convert equations of circles and ellipses to standard form, integrate rational functions, and solve differential equations. It is a foundational algebraic skill that appears throughout advanced mathematics.
Frequently Asked Questions
Why complete the square instead of using the quadratic formula?
Completing the square reveals the vertex and axis of symmetry directly, which is useful for graphing. The quadratic formula is derived from completing the square and is faster for finding roots only.
What is vertex form of a parabola?
Vertex form is y = a(x − h)² + k, where (h, k) is the vertex of the parabola. When a > 0, the vertex is the minimum; when a < 0, it is the maximum.
How do I complete the square when a ≠ 1?
Factor out a first: a(x² + (b/a)x) + c. Then complete the square inside the brackets, remembering to multiply the added square by a when moving it outside.
Does completing the square always work?
Yes. Any quadratic with real coefficients can be written in vertex form. The vertex form makes it easy to see whether the quadratic has 0, 1, or 2 real roots based on the sign of k.
Is this calculator free?
Yes, completely free with no sign-up required.