Mathematics, Statistics & Geometry

Completing the Square Calculator

Convert quadratic equations to vertex form instantly using completing the square. Find the exact vertex coordinates (h,k) with clear steps.

Vertex Form
2(x - 2.0000)² + 7.0000 = 0
Vertex Coordinate (h, k)(-2.0000, 7.0000)
Calculation StepsEquation: 2x² + 8x + 15 = 0 1. Factor 'a' from the x terms: 2(x² + 4.0000x) + 15 = 0 2. Find the value to complete the square inside parentheses: (b/(2a))² = (8/(2*2))² = 4.0000 3. Add and subtract this value: h = -b/(2a) = -2.0000 k = c - a*h² = 15 - 2*(-2.0000)² = 7.0000 4. Final Vertex Form: 2(x - 2.0000)² + 7.0000 = 0

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Master Quadratic Transformations

The Completing the Square Calculator is a lifesaver for algebra students and engineers. While the quadratic formula is great for finding roots, completing the square is the only way to convert an equation into Vertex Form, revealing the exact geometric center of the parabola.

a(x−h)2+k=0\begin{aligned} a(x - h)^2 + k = 0 \end{aligned}

Where:
a=
The scaling factor of the parabola
h=
The x-coordinate of the parabola's vertex (-b/2a)
k=
The y-coordinate of the parabola's vertex

The Step-by-Step Technique

Completing the square manually can be highly error-prone due to the fraction arithmetic involved. The process follows these core steps:

  1. Factor: Pull the leading coefficient (a) out of the first two terms.
  2. Halve and Square: Take the new x-coefficient, divide it by 2, and square it.
  3. Add and Subtract: Add this new value inside the parentheses to create a perfect square trinomial, and subtract the equivalent value outside to keep the equation balanced.
  4. Simplify: Factor the perfect square and simplify the constants.

Why is this important?

  • Physics (Projectile Motion): Finding the maximum height of an object thrown in the air (the y-coordinate of the vertex).
  • Optimization Problems: Determining the exact x-value that produces the minimum cost or maximum profit in a quadratic business model.
  • Pre-Calculus: It is a required skill for graphing conic sections (circles, ellipses, hyperbolas, and parabolas).

Frequently Asked Questions

It is an algebraic technique used to convert a quadratic equation from standard form (ax² + bx + c) into vertex form (a(x - h)² + k).

Vertex form instantly reveals the peak or valley (the vertex) of the parabola, making it incredibly easy to graph. The vertex is located at the coordinates (h, k).

The x-coordinate of the vertex (h) is found using the formula -b / (2a). The y-coordinate (k) is found by plugging 'h' back into the original equation, or using c - (b²/4a).

Yes! Once in vertex form, you can easily solve for x by isolating the squared term and taking the square root of both sides. This is actually how the Quadratic Formula is derived.

Yes, but you must first factor 'a' out of the x² and x terms before completing the square inside the parentheses. Our calculator handles this step automatically.

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