Completing the Square
Rewrite a quadratic in vertex form, step by step
About this calculator
The Completing the Square calculator rewrites any quadratic ax² + bx + c into vertex form a(x − h)² + k, showing every step: factoring out a, halving the x-coefficient, squaring it, and grouping the perfect square. It also gives the vertex, axis of symmetry, minimum or maximum value, and real roots, with a parabola visual. Useful for algebra homework, graphing, and deriving the quadratic formula.
How to use the Completing the Square calculator
- Enter the values for your problem into the input fields.
- Read the result — it updates instantly as you type.
- Check the formula and the visual explanation to follow how the answer was found.
- Copy the page URL to share the exact calculation.
Common examples
- x² − 6x + 5 → (x − 3)² − 4, vertex (3, −4), roots x = 1 and x = 5
- x² + 6x + 11 → (x + 3)² + 2, vertex (−3, 2), no real roots
- 2x² − 8x + 3 → 2(x − 2)² − 5, vertex (2, −5)
- x² + 4x + 4 → (x + 2)², a perfect square with double root x = −2
- −x² + 2x + 3 → −(x − 1)² + 4, vertex (1, 4), opens downward
Frequently asked questions
How does the Completing the Square calculator work?
Enter your values and the calculator applies the exact mathematical method for this problem, showing the result together with the formula it used. Everything is computed with high-precision arithmetic, so the answer you see is not limited by ordinary floating-point rounding.
When would I use the Completing the Square calculator?
It is useful for homework and exam preparation, for checking work done by hand, and whenever a step in a larger problem needs this calculation done quickly and reliably.
How accurate are the results?
Calculations run at 30 significant digits of precision internally. Displayed values are rounded for readability, but the underlying result is far more precise than a typical hand or pocket-calculator computation.
Can I share a specific calculation with someone else?
Yes. Every value you enter updates the URL, so copying the address bar and sharing the link reopens the exact same calculation for whoever clicks it.