Completing the Square Calculator
Calculate the completed-square form, vertex coordinates, axis of symmetry, discriminant, and roots based on coefficients a, b, and c. Completing the Square Calculator helps students, teachers, and algebra learners rewrite quadratic equations and understand their graphs.
Highlights
- Rewrite quadratic expressions in completed-square (vertex) form.
- Calculate the parabola’s vertex coordinates.
- Identify the axis of symmetry.
- Find the discriminant and determine solution types.
- Solve for real or complex roots of quadratic equations.
Table of Contents
How Completing the Square Calculator Works?
The calculator starts with the quadratic expression ax2 + bx + c, where a must be nonzero. It rewrites the expression in vertex form, a(x − h)2 + k, by factoring out a from the quadratic and linear terms, adding and subtracting the square of half the adjusted linear coefficient, and then simplifying.
Comparing this result with a(x − h)2 + k gives the vertex coordinates and the axis of symmetry.
Thus, the vertex is (h, k), and the vertical axis of symmetry is x = h. The calculator also evaluates the discriminant, which determines the nature of the roots.
| Discriminant | Roots | Graph interpretation |
|---|---|---|
| D > 0 | Two distinct real roots | The parabola crosses the x-axis twice |
| D = 0 | One repeated real root | The parabola touches the x-axis at its vertex |
| D < 0 | Two complex-conjugate roots | The parabola has no real x-intercepts |
When roots are requested, they are calculated using the quadratic formula:
The inputs a, b, and c are unitless decimal coefficients supported within the calculator's stated input range.
Practical Use Cases
Algebra students: Enter the coefficients of a quadratic to check work when rewriting an expression in completed-square form, locating the vertex, identifying the axis of symmetry, and solving for roots.
Teachers and tutors: Use the calculator to create quick examples that compare standard form and vertex form, demonstrate how the discriminant affects the graph, and verify classroom exercises.
Graphing and function learners: Use the vertex coordinates and axis of symmetry to understand whether a parabola opens upward or downward, locate its turning point, and predict its x-intercepts.
Students studying equations: Use the discriminant and roots output to distinguish between two real solutions, one repeated solution, and a pair of complex solutions without manually repeating every algebraic step.
Accuracy and Limitations
The calculator applies the standard completed-square identities, vertex formulas, discriminant formula, and quadratic formula, so results are mathematically exact for valid inputs; displayed decimal results may be rounded by the interface. The leading coefficient a must be nonzero, and each coefficient must remain within the supported range of −100,000 to 100,000 with decimal entry increments of 0.01. Very large or finely balanced decimal coefficients can produce ordinary floating-point rounding in displayed values, and a negative discriminant produces complex roots rather than real x-intercepts.
Quick Examples
Example #1: Perfect Square Quadratic
Rewrite a quadratic expression that produces a perfect-square binomial and a repeated real root.
Example #2: Quadratic with Two Real Roots
Analyze a quadratic with a positive discriminant to identify two distinct real roots and its vertex.
Example #3: Quadratic with Complex Roots
Complete the square for a quadratic with a negative discriminant and no real x-intercepts.
