Completing the Square Calculator
Convert ax² + bx + c to vertex form a(x - h)² + k by completing the square, and find the vertex and roots.
This free online completing the square calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.
Coefficient of x² (must not be zero)
Coefficient of x
Constant term
Results
h (x-shift)
0
k (y-shift / minimum or maximum)
0
b/(2a)
0
Discriminant
0
How to Use This Calculator
Enter your input values
Fill in all required input fields for the Completing the Square Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.
Review your inputs
Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.
Read the results
The Completing the Square Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.
Explore parameter sensitivity
Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.
When to Use This Calculator
- •Use the Completing the Square Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
- •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
- •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
- •Use it to explore the behavior of mathematical functions across a range of inputs.
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About Completing the Square Calculator
The Completing the Square Calculator transforms a quadratic expression ax² + bx + c into its vertex form a(x - h)² + k. This technique is one of the most versatile tools in algebra, used not only to solve quadratic equations but also to derive the quadratic formula itself, convert equations of conic sections to standard form, and evaluate integrals in calculus. The vertex form immediately reveals the vertex (h, k) of the parabola, its axis of symmetry x = h, and whether it opens upward (a > 0) or downward (a < 0). The value k represents the minimum (when a > 0) or maximum (when a < 0) of the quadratic function. Completing the square is essential for graphing parabolas, solving optimization problems, and understanding transformations of functions.
The Math Behind It
Formula Reference
Vertex Form
a(x - h)² + k
Variables: h = -b/(2a), k = c - b²/(4a)
Completing the Square Identity
x² + bx = (x + b/2)² - (b/2)²
Variables: Add and subtract (b/2)² to create a perfect square
Worked Examples
Example 1: Monic quadratic
Complete the square for x² + 6x + 2
(x + 3)² - 7
Example 2: Non-monic quadratic
Complete the square for 2x² - 8x + 5
2(x - 2)² - 3, vertex at (2, -3)
Common Mistakes & Tips
- !Forgetting to factor out the leading coefficient a before completing the square
- !Adding (b/2)² inside the parentheses but forgetting to subtract a·(b/2)² outside (when a was factored out)
- !Confusing the sign of h: vertex form is a(x - h)², so x² + 6x gives (x + 3)² meaning h = -3
- !Not dividing b by a first when a ≠ 1
Related Concepts
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Frequently Asked Questions
Why is completing the square useful when we have the quadratic formula?
Completing the square gives the vertex form, which is essential for graphing and optimization. The quadratic formula only finds roots. Additionally, completing the square is used in calculus for integration and in converting conic section equations.
Does completing the square work for all quadratics?
Yes, every quadratic expression can be written in vertex form using completing the square. The process always works, regardless of whether the roots are real or complex.
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