Quadratic Formula Calculator
Solve ax² + bx + c = 0: real or complex roots, discriminant, vertex and axis of symmetry.
In short
Solve ax² + bx + c = 0: real or complex roots, discriminant, vertex and axis of symmetry.
Formula: For ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) ÷ 2a. Discriminant D = b² − 4ac; vertex h = −b ÷ 2a, k = a h² + b h + c.
What this calculator does
Solve ax² + bx + c = 0: real or complex roots, discriminant, vertex and axis of symmetry. Formula, worked example, sources, FAQs and limitations included.
You enter coefficient a (x²), coefficient b (x) and constant c, and the quadratic formula calculator shows the answer along with the working, so you can check it or try other numbers.
The page shows the formula, a numeric worked example, and the assumptions that affect this math calculation.
Inputs and what they mean
- Coefficient a (x²)
- — number value.
- Coefficient b (x)
- — number value.
- Constant c
- — number value.
How to use it
- Enter each value in the unit shown next to the box (use the unit converter first if your numbers are in other units).
- Check the breakdown to see every intermediate step.
- Read the limitations before relying on the result.
Formula
Source: OpenStax College Algebra 2e §2.5, Quadratic Equations.
D > 0 gives two real roots, D = 0 one repeated root, D < 0 two complex conjugate roots.
For ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) ÷ 2a. Discriminant D = b² − 4ac; vertex h = −b ÷ 2a, k = a h² + b h + c.
Inputs used: Coefficient a (x²), Coefficient b (x), Constant c.
Worked example
x² − 5x + 6 = 0
- D = 25 − 24 = 1.
- x = (5 ± 1) ÷ 2 → 3 and 2.
- Vertex h = 2.5, k = −0.25.
Reading the result
The headline figure is the main answer. Any breakdown underneath shows the parts that make it up, so you can check the working and see what changes when you adjust an input.
Limitations and assumptions
- Real-number coefficients only.
- Displayed roots are rounded; very large or tiny coefficients can lose floating-point precision.
- Exact mathematical relationships, rounded for display. Check units and rounding before using a result in graded, engineering or financial work.
- The result depends on the values you enter for this quadratic formula calculator; it does not supply missing rates, rules, prices, dates, or assumptions for you.
Reference: OpenStax
Last reviewed:
Common questions
What does a negative discriminant mean?
There are no real roots; the calculator shows the two complex conjugate roots.
Why can't a be zero?
Without the x² term the equation is linear and the quadratic formula divides by zero.
What is the vertex?
The turning point of the parabola y = ax² + bx + c.
How do I use the Quadratic Formula Calculator?
Enter the required values for Coefficient a (x²), Coefficient b (x), Constant c. The calculator applies the formula on this page and shows the main result with any supporting breakdown so you can check the arithmetic.
What formula does the Quadratic Formula Calculator use?
For ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) ÷ 2a. Discriminant D = b² − 4ac; vertex h = −b ÷ 2a, k = a h² + b h + c. The visible formula section above lists the calculation path and the edge cases the page handles, so the result can be checked without relying on the form alone.
Can the Quadratic Formula Calculator be used for exact decisions?
Use it as a calculation aid, not as a substitute for checking the underlying rule, contract, policy, or professional advice that applies to your situation. When a result depends on local rules, personal details, prices, or dates, enter those values yourself and confirm them before acting.
Sources
Method based on OpenStax.
Cite this page
Free Calculator. (2026, November 3). Quadratic Formula Calculator. https://freecalculator.tech/tools/quadratic-formula-calculator
"Quadratic Formula Calculator." Free Calculator, 3 Nov. 2026, https://freecalculator.tech/tools/quadratic-formula-calculator.
Quadratic Formula Calculator — Free Calculator, https://freecalculator.tech/tools/quadratic-formula-calculator
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