What Is a Completing the Square Calculator?
A Completing the Square Calculator is a simple online tool that helps you convert any Quadratic expression of the form ax² + bx + c into its completed square form. Completing the square is a key algebraic method used for solving quadratic equations, finding vertex form, analysing graphs, and understanding the minimum or maximum point of a parabola.
Instead of doing long manual steps, this calculator instantly shows:
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The perfect square form
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Step-by-step working
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Values of h and k
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The final expression: a(x + h)² + k
This makes it perfect for students, teachers, and anyone working with quadratic expressions.
For more helpful tools and smart calculation guides, explore our full range of online calculators in Ireland.
How to Use the Completing the Square Calculator
Using this tool is very easy:
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Enter the value of “a” – the coefficient of x²
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Enter the value of “b” – the coefficient of x
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Enter the value of “c” – the constant term
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Click the Calculate button
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The tool will instantly show:
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The completed square form
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Every step used to derive it
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A simplified and easy-to-read solution
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You can use this calculator as many times as needed, whether for homework, exams, algebra practice, or general learning.
You can also go through - Simpson’s Rule Calculator
Why Complete the Square?
Completing the square helps you:
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Convert standard form to vertex form
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Solve quadratic equations
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Understand the turning point of a parabola
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Rewrite expressions for integration
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Analyse function transformations
It is one of the most important skills in secondary-level algebra and appears in many mathematics exams.
Formula for Completing the Square
For any quadratic:
ax² + bx + c
Step 1: Factor out “a”
Step 2: Add and subtract (b/2a)² inside the bracket
Step 3: Rewrite as a perfect square
Step 4: Simplify the constant term outside the square
Final form becomes:
a(x + h)² + k
Where:
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h = b / (2a)
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k = c − (b² / 4a)
Example of Completing the Square
Let’s complete the square for:
2x² + 8x + 5
Step 1: Factor out 2
→ 2(x² + 4x) + 5
Step 2: Add & subtract (4/2)² = 4
→ 2[(x² + 4x + 4 − 4)] + 5
Step 3: Rewrite perfect square
→ 2[(x + 2)² − 4] + 5
Step 4: Simplify
→ 2(x + 2)² − 8 + 5
→ 2(x + 2)² − 3
Final Answer:
2(x + 2)² – 3
You can check this same example using the calculator above.
Features of This Calculator
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Fast and accurate
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Step-by-step breakdown
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Supports decimals & negative numbers
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Mobile-friendly
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Ideal for school and college maths
Frequently Asked Questions (FAQ)
1. What is completing the square used for?
It is mainly used to write a quadratic in vertex form, solve equations, or find the turning point of a parabola.
2. Why is the method important?
It gives insight into the shape and behaviour of quadratic graphs and is widely used in mathematics and physics.
3. Does the calculator support negative numbers?
Yes, the tool accepts positive, negative, and decimal coefficients.
4. Can completing the square solve quadratic equations?
Yes. Once rewritten in the form a(x + h)² + k = 0, solving becomes very easy.
5. Do I need to know the formula?
No. The calculator does all the steps automatically and shows the full solution.