Completing the Square Calculator
Convert ax^2 + bx + c to vertex form a(x - h)^2 + k and find the vertex.
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Request a ToolHow to Use the Completing the Square Calculator
This calculator converts a quadratic from standard form to vertex form by completing the square.
- Enter coefficients a, b, and c. These define the quadratic ax^2 + bx + c.
- Read the vertex form. The result shows a(x - h)^2 + k along with the vertex coordinates (h, k).
- Copy or share. Use the buttons to copy or create a shareable link.
About Completing the Square
Completing the square is an algebraic technique that rewrites ax^2 + bx + c as a(x - h)^2 + k. The vertex of the parabola is at (h, k), where h = -b/(2a) and k = c - b^2/(4a). This form immediately reveals the vertex, the axis of symmetry (x = h), and whether the parabola opens up (a > 0) or down (a < 0). The technique is also used to derive the quadratic formula itself.
Frequently Asked Questions
What is completing the square used for?
It converts a quadratic to vertex form, making it easy to identify the vertex, axis of symmetry, and maximum or minimum value. It is also used to derive the quadratic formula and to solve certain integrals.
How do you find h and k?
h = -b / (2a) and k = c - b^2 / (4a). These give the vertex (h, k) of the parabola.
Does completing the square work when a is not 1?
Yes. Factor out a from the first two terms, complete the square inside, then distribute a back. This calculator handles any nonzero value of a.