# Completing the Square Calculator

Complete the square for any quadratic ax² + bx + c. Convert to vertex form, find the vertex coordinates, roots, and discriminant.

## What this calculates

Convert any quadratic expression from standard form ax² + bx + c to vertex form a(x - h)² + k. This calculator finds the vertex, discriminant, and roots of the quadratic equation.

## Inputs

- **a (coefficient of x²)** — The coefficient of the x² term. Must not be zero.
- **b (coefficient of x)** — The coefficient of the x term.
- **c (constant)** — The constant term.

## Outputs

- **Vertex Form** — formatted as text — The quadratic expressed in vertex form a(x - h)² + k.
- **Vertex x (h)** — The x-coordinate of the vertex.
- **Vertex y (k)** — The y-coordinate of the vertex.
- **Roots** — formatted as text — The real roots of the equation ax² + bx + c = 0, if they exist.
- **Discriminant** — b² - 4ac. Positive = two real roots, zero = one repeated root, negative = no real roots.

## Details

Completing the square is a technique that rewrites a quadratic expression into vertex form, which makes it easy to read off the vertex of the parabola, determine its direction of opening, and find its roots. The vertex form a(x - h)² + k reveals that the parabola has its vertex at (h, k), opens upward when a > 0, and opens downward when a < 0.

The process works by taking ax² + bx + c, factoring out a from the first two terms, and adding and subtracting the square of half the linear coefficient inside the parentheses. This yields a = coefficient, h = -b/(2a) for the x-coordinate of the vertex, and k = c - b²/(4a) for the y-coordinate.

Completing the square is also the derivation behind the quadratic formula. By applying the technique to the general equation ax² + bx + c = 0, you arrive at x = (-b ± sqrt(b² - 4ac)) / (2a). The discriminant b² - 4ac determines the nature of the roots: positive gives two distinct real roots, zero gives one repeated root, and negative gives two complex conjugate roots.

## Frequently Asked Questions

**Q: What is completing the square?**

A: Completing the square is an algebraic technique that rewrites a quadratic expression ax² + bx + c into the equivalent vertex form a(x - h)² + k, where h = -b/(2a) and k = c - b²/(4a). It reveals the vertex of the parabola and simplifies finding roots.

**Q: Why is vertex form useful?**

A: Vertex form a(x - h)² + k immediately tells you the vertex of the parabola is at (h, k), whether it opens up (a > 0) or down (a < 0), and the axis of symmetry is x = h. This makes graphing much easier than working from standard form.

**Q: How does the discriminant relate to the roots?**

A: The discriminant is b² - 4ac. If it is positive, the quadratic has two distinct real roots. If it is zero, there is exactly one repeated real root. If it is negative, there are no real roots, only two complex conjugate roots.

**Q: Can I complete the square when a is not 1?**

A: Yes. First factor out a from the x² and x terms: a(x² + (b/a)x) + c. Then complete the square inside the parentheses by adding and subtracting (b/(2a))². The result is a(x + b/(2a))² + c - b²/(4a), which is the vertex form.

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Source: https://vastcalc.com/calculators/math/completing-the-square
Category: Math
Last updated: 2026-04-21