Last updated October 12, 2026 · Report an error
Completing the square is the algebraic technique behind both the quadratic formula and vertex form — it rewrites ax²+bx+c as a perfect square plus a constant. This tool walks through every step of that derivation rather than jumping straight to the h/k formulas.
How it works
The x² and x terms are factored so the leading coefficient a is pulled out front, then half the resulting x-coefficient is squared and both added and subtracted inside — turning the first three terms into a perfect square trinomial that factors neatly.
- Enter a (coefficient of x²).
- Enter b (coefficient of x).
- Enter c (constant).
- Click Calculate to see your results.
Examples
A simple case
x²+6x+5 becomes (x+3)²-9+5 = (x+3)²-4.
With a leading coefficient
2x²-8x+3 becomes 2(x-2)²-8+3 = 2(x-2)²-5.
Who should use it
- Algebra coursework introducing completing the square before the quadratic formula.
- Understanding where the vertex form shortcut formulas actually come from.
Industry applications
- Mathematics education
Advantages
- Shows the full algebraic derivation, not just the final formula.
- Works for any leading coefficient, not just a=1.
Limitations
- Only handles quadratics (degree 2) — cubics and higher require different techniques entirely.
Common mistakes to avoid
- Forgetting to multiply the subtracted square term by the leading coefficient a before moving it outside the parentheses.
Best practices
- Factor out the leading coefficient from only the x² and x terms first — leave the constant term outside.
Tips
- If a ≠ 1, remember to factor it out of both the x² and x terms before completing the square — a common place students lose track of a factor.
Frequently asked questions
The Vertex Form Calculator applies the h=-b/2a and k=c-b²/4a shortcut formulas directly, while this tool shows the underlying factor-and-complete-the-square derivation those formulas come from.
Adding and subtracting the identical amount changes nothing about the expression's value, but it lets the first three terms be regrouped into a perfect square trinomial.