Complete the square for any quadratic ax²+bx+c. Get vertex form, vertex coordinates, axis of symmetry, discriminant, roots, and step-by-step solution.
Completing the square is one of the most important algebraic techniques for working with quadratic expressions. Given a quadratic in standard form ax² + bx + c, completing the square rewrites it in vertex form a(x − h)² + k, where (h, k) is the vertex of the parabola. This transformation reveals the minimum or maximum value of the quadratic, the axis of symmetry, and provides a direct path to finding the roots.
The technique works by taking the coefficient of x, halving it, squaring the result, and adding and subtracting this value to create a perfect square trinomial. This calculator automates the entire process, showing you each step along the way. It computes the vertex coordinates, axis of symmetry, discriminant, roots (real or complex), and y-intercept.
Completing the square is not just a classroom exercise — it is essential in calculus for integrating rational functions, in physics for analyzing projectile motion, in statistics for deriving the normal distribution, and in optimization problems throughout engineering. Whether you need to quickly convert a quadratic to vertex form for graphing or want to understand the step-by-step algebra, this calculator handles it all with clear explanations.
This completing the square calculator reduces manual rework when you need quick checks for assignments, exam prep, and design calculations. You can enter a (leading coefficient), b (linear coefficient), c (constant) and immediately see dependent measurements, validity checks, and geometry relationships in one place. That makes it easier to catch input mistakes early and confirm your final answer before moving to the next step.
Given ax² + bx + c: h = −b / (2a) k = c − b² / (4a) Vertex form: a(x − h)² + k Discriminant: Δ = b² − 4ac Roots: x = (−b ± √Δ) / (2a)
Result: For a=1, b=6, c=5, the tool returns the solved completing the square outputs shown in the result cards.
This example uses a realistic input set from the calculator workflow. After entry, the calculator applies the built-in completing the square formulas and reports derived values, checks, and classifications automatically.
This page is tailored to completing the square, with outputs tied directly to the form fields (a (leading coefficient), b (linear coefficient), c (constant)). Instead of a one-line formula dump, it consolidates validation, derived metrics, and interpretation so you can solve and verify in one pass.
Use this tool for homework checks, worksheet generation, tutoring walkthroughs, and quick engineering geometry estimates. Presets and visual output blocks make it easier to compare scenarios and understand how each input affects the final result.
Keep units consistent, match each value to the correct field, and watch validity indicators before using the final numbers. If your case looks off, change one input at a time and use the output details to identify the mismatch quickly.
It means rewriting ax² + bx + c as a(x − h)² + k by creating a perfect square trinomial. This reveals the vertex (h, k) of the parabola.
It converts a quadratic to vertex form, making it easy to identify the vertex, axis of symmetry, and whether the parabola opens up or down. It also derives the quadratic formula.
Factor a out of the x² and bx terms: a(x² + (b/a)x) + c, then complete the square inside the parentheses. The calculator handles this automatically.
Standard form ax² + bx + c shows coefficients directly. Vertex form a(x − h)² + k shows the vertex directly. They represent the same quadratic.
The quadratic has no real roots — its parabola does not cross the x-axis. The roots are complex conjugates of the form (−b ± i√|Δ|) / (2a).
Yes, completing the square works for any quadratic expression with a ≠ 0. The leading coefficient can be any nonzero real number.