Math

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Completing the Square Calculator.

Rewrite a quadratic in vertex form and show its discriminant.

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Set your values

Results update as you type.

Result: vertexX: 2.00000000 · vertexY: -1.00000000 · discriminant: 4.00000000 · form: 1(x − 2)² − 1

Result

vertexX: 2.00000000 · vertexY: -1.00000000 · discriminant: 4.00000000 · form: 1(x − 2)² − 1

Use Cases

Graphing Quadratics

Use the vertex form to quickly identify the vertex and axis of symmetry, making it easier to sketch the parabola accurately.

Example: For y = x² - 4x + 5, the calculator gives (x - 2)² + 1, so the vertex is (2, 1).

Solving Optimization Problems

Find the maximum or minimum value of a quadratic function by converting to vertex form, which reveals the optimal point directly.

Example: Maximize profit P = -2x² + 12x - 10: vertex form gives -2(x - 3)² + 8, so max profit is 8 at x = 3.

Frequently Asked Questions

What does the completing the square calculator do?
This calculator takes the coefficients a, b, and c of a quadratic equation ax² + bx + c and rewrites it in vertex form a(x - h)² + k. It shows the steps to complete the square, making it easy to find the vertex and understand the transformation.
How do I use the calculator?
Simply enter the values for a, b, and c from your quadratic equation. The calculator will compute the vertex form and display the result. Ensure that 'a' is not zero, as that would make it a linear equation, not quadratic.
What is vertex form and why is it useful?
Vertex form is a(x - h)² + k, where (h, k) is the vertex of the parabola. It's useful because it directly shows the maximum or minimum point of the quadratic, making graphing and solving optimization problems easier.

Tips & Common Mistakes

Tips

  • Ensure 'a' is not zero; otherwise, the equation is not quadratic.
  • Double-check your input values for a, b, and c to avoid errors in the vertex form.
  • Use the vertex form to quickly find the vertex coordinates (h, k) for graphing.
  • If 'a' is negative, the parabola opens downward, and the vertex represents the maximum point.

Common Mistakes to Avoid

  • Forgetting to factor out 'a' from the x² and x terms before completing the square.
  • Incorrectly calculating the value to add and subtract inside the parentheses (should be (b/2a)²).
  • Misinterpreting the sign of h in the vertex form: the vertex is (h, k), not (-h, k).

Last updated: August 13, 2026