Quadratic Formula Calculator — Solve ax² + bx + c = 0
Solve any quadratic equation with the quadratic formula. Find both roots, discriminant, vertex, axis of symmetry, and factored form with step-by-step solutions.
Completing the Square Calculator
Complete the square for any quadratic ax²+bx+c. Get vertex form, vertex coordinates, axis of symmetry, discriminant, roots, and step-by-step solution.
Enter coefficients of ax² + bx + c
Planning notes, formulas, and examples
About the Completing the Square Calculator
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.
When This Page Helps
Use this when you need the completed-square form of a quadratic for solving, graphing, or converting to vertex form without reworking every algebra step manually. It is helpful for homework, teaching, and equation checks because the original coefficients, completed form, and verification stay connected.
How to Use the Inputs
- Enter the coefficients a, b, and c of your quadratic ax² + bx + c.
- Or click a preset to load a common quadratic expression.
- The coefficient a must not be zero (otherwise it is not a quadratic).
- Review the vertex form, vertex, axis of symmetry, and roots.
- Expand the step-by-step solution to see each algebraic manipulation.
- Check the reference table comparing different quadratic forms.
Formula used
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)Example Calculation
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.
Tips & Best Practices
- If a ≠ 1, factor it out from the x² and x terms first before completing the square.
- The vertex is always at x = −b/(2a) regardless of the method used.
- A positive discriminant means two real roots; zero means a double root; negative means complex roots.
- Completing the square is the algebraic basis behind the quadratic formula itself.
- For graphing, vertex form immediately gives you the turning point and direction of opening.
When To Use This Calculator
Complete the square for any quadratic ax²+bx+c. Get vertex form, vertex coordinates, axis of symmetry, discriminant, roots, and step-by-step solution. Use it when you need a repeatable calculation in the math / geometry category and want the setup, result, and supporting values kept together. This is especially helpful when small input changes, unit choices, or rounding decisions can change the final number.
How To Check The Result
Start by confirming that the inputs match the formula shown on the page. Then compare the main output with the worked example and any secondary values shown by the calculator. If the result will be used in another calculation, keep extra precision until the final step and record the assumptions beside the number.
Practical Notes
Treat the result as a calculation aid rather than a substitute for context. For schoolwork, include the formula and substitution steps. For planning, technical, financial, or health-related decisions, verify important numbers against primary records, current rules, or a qualified professional before acting on them.
Sources & Methodology
Last updated:
Frequently Asked Questions
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.
More in this topic
Completing the Square Practice
Practice completing the square with generated problems. Enter your answer for h and k, check if correct, and get step-by-step solutions with multiple difficulty levels.
Why Completing the Square Works
Understand WHY completing the square works through geometric area interpretation. See algebraic steps alongside visual area models, comparison bars, and step breakdowns.