Calculadora de Completar o Quadrado
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
ax² + bx + c = a(x − h)² + k com h = −b/(2a), k = c − b²/(4a)
About this calculator
The Completing the Square Calculator converts a quadratic expression from ax² + bx + c to the canonical form a(x − h)² + k. This process is essential in algebra for simplifying equations, solving optimization problems, and understanding parabola properties. The tool automatically computes h and k using the formulas h = −b/(2a) and k = c − b²/(4a), displaying the step-by-step result.
To use the calculator, enter the coefficients a, b, and c of the quadratic equation. The result shows the expression as a(x − h)² + k, highlighting the values of h and k. This is useful for finding the vertex of a parabola, solving equations by completing the square, or simplifying integrals. The calculator accepts real numbers, including fractions and decimals.
Use cases include solving quadratic equations that cannot be easily factored, finding the maximum or minimum value of quadratic functions, and converting equations for graphical analysis. For example, in projectile motion, the canonical form reveals the highest point of the trajectory. The tool also assists in algebra proofs and exercises.
Cautions: ensure the coefficient a is nonzero. The calculator assumes the expression is quadratic. For equations with fractional coefficients, use the correct decimal or fractional representation. Remember that the canonical form is equivalent to the original but highlights the parabola's vertex.
Frequently asked questions
What does completing the square mean?
Completing the square is a method to rewrite a quadratic expression ax² + bx + c as a(x − h)² + k, making it easier to find the vertex and solve equations.
How do I calculate h and k manually?
h = −b/(2a) and k = c − b²/(4a). The calculator does this automatically, but you can check the steps shown.
Does the calculator work with fractional coefficients?
Yes, enter fractions as decimals or use fraction notation (e.g., 1/2). The calculator handles real numbers.
What is the use of the canonical form?
It shows the parabola's vertex (h, k), which is useful for graphing, optimization, and solving quadratic equations.
What if a is zero?
If a = 0, the expression is not quadratic. The calculator may not work correctly; make sure a ≠ 0.