🔳Completing the Square Calculator
Convert ax² + bx + c to vertex form a(x − h)² + k with every step shown.
Free Completing the Square Calculator
Completing the square is one of the trickier algebra techniques to follow by hand. Our free completing the square calculator takes any quadratic ax² + bx + c and converts it to vertex form a(x − h)² + k, showing every algebraic step along the way.
Enter your coefficients and watch the process unfold: factoring out the leading coefficient, halving and squaring the linear term, and rewriting the expression as a perfect square plus a constant. From there, the calculator derives the vertex, the axis of symmetry, and the roots — cross-checked against the quadratic formula.
Cases with no real roots are explained clearly rather than shown as an error, and coefficients are handled exactly, including clean fractions rather than rounded decimals. No sign-up, no downloads, no limits.
About the Completing the Square Calculator
Features
Step-by-step derivation
Every algebraic move shown in order, not just the final vertex form.
Vertex, axis, roots
All three derived directly from the completed-square form.
Fractional precision
Clean fractions shown exactly instead of rounded decimals.
No-real-root handling
Complex-root cases explained clearly, never left as an error.
Frequently Asked Questions
How does the completing the square calculator work?
Enter the coefficients of your quadratic and it converts it to vertex form, showing every algebraic step.
Is it free?
Completely free, no sign-up, downloads, or limits.
What is vertex form?
Writing a quadratic as a(x - h)² + k, where (h, k) is the vertex of the parabola.
Does it show every step?
Yes, factoring out the leading coefficient, halving and squaring the linear term, and rewriting as a perfect square are all shown in order.
Can it find the vertex and axis of symmetry?
Yes, both are derived directly from the completed-square form.
Can it find the roots too?
Yes, the roots are solved from the vertex form and cross-checked against the quadratic formula.
What if there are no real roots?
The calculator explains clearly that the roots are complex/non-real rather than showing an error.
Does it handle fractional coefficients?
Yes, results are shown as exact fractions where possible, not rounded decimals.
Is my data stored?
No; everything runs in your browser and isn't stored or sent.
Does it work on mobile?
Yes, fully responsive on any screen size.