About the Completing the Square Calculator
Completing the Square Calculator rewrites a quadratic expression ax² + bx + c in vertex form, a(x − h)² + k. Enter a, b and c and the page shows the steps: factor out a, take half of the coefficient of x, square it, add and subtract it, and write the perfect square. It then gives the vertex (h, k) of the parabola, the axis of symmetry, whether the vertex is a minimum or a maximum and the roots, including complex roots when the parabola does not cross the x-axis. It is useful for algebra, for graphing parabolas and for deriving the quadratic formula.
Updated October 2026
How to use the Completing the Square Calculator
- 1Enter a, the number in front of x².
- 2Enter b, the number in front of x, and c, the constant.
- 3Read the vertex form and the steps.
- 4Read the vertex, axis of symmetry and roots.
- 5Copy the result.
Why complete the square
A quadratic in standard form hides where its turning point is. Completing the square reveals it: the vertex form a(x − h)² + k shows that the lowest or highest point is at x = h with value k. The same trick solves any quadratic equation and proves the quadratic formula.
A worked example
Take x² + 6x + 5. Half of 6 is 3 and 3² is 9. Write x² + 6x + 9 − 9 + 5 = (x + 3)² − 4. The vertex is (−3, −4), the minimum value is −4 and the roots are x = −1 and x = −5, since (x + 3)² = 4 means x + 3 = ±2.
- Half of b, squared: (b/2)²
- Add and subtract it so the value does not change
- Write the first three terms as a square
When a is not 1
Factor a out of the first two terms first. For 2x² − 8x + 3, that gives 2(x² − 4x) + 3, then 2(x − 2)² − 8 + 3 = 2(x − 2)² − 5. The calculator shows the steps so you can follow each line.
Related tools
Check the roots with the Quadratic Equation Solver, multiply brackets with the FOIL Calculator and see more of the parabola with the Parabola Calculator. The Graphing Calculator draws the curve.
Features
- Vertex form a(x − h)² + k
- Step-by-step working
- Vertex and axis of symmetry
- Minimum or maximum value
- Real and complex roots
- Private: nothing is uploaded
Frequently asked questions
What does completing the square mean?
Rewriting a quadratic so that part of it is a perfect square, (x + p)². It turns ax² + bx + c into a(x − h)² + k.
What are the steps?
Factor a out of the x² and x terms, add and subtract (b/2a)² inside the bracket, write the bracket as a square and combine the constants.
How do I find the vertex?
In vertex form a(x − h)² + k, the vertex is the point (h, k). The calculator gives it directly.
Is the vertex a minimum or a maximum?
If a is positive the parabola opens upwards and the vertex is a minimum. If a is negative it opens downwards and the vertex is a maximum.
How do I solve a quadratic by completing the square?
Set a(x − h)² + k = 0, move k across, divide by a, take square roots of both sides and add h. The calculator also lists the roots.
What if the roots are complex?
If k and a have the same sign, the parabola never touches the x-axis, so the roots are complex numbers a ± bi, which the page writes out.
Why can a not be 0?
Then there is no x² term and the expression is not a quadratic.
Is my input stored?
No. It is calculated in your browser.