CalcCommonsHow to use
Enter signed coefficients exactly as they appear after moving every term to the left side. Include zero for a missing b or c term. Calculate to see both roots, and use the curve as a visual explanation, not a scale drawing.
For related tasks, try the Factoring Calculator.
Formula & how it works
D = b² − 4ac; x = (−b ± √D) ÷ (2a). Vertex = (−b/(2a), −D/(4a)).
Worked examples
Example 1
For x² − 5x + 6 = 0, D = 25 − 24 = 1. The roots are 2 and 3; the vertex is (2.5, −0.25).
Example 2
For x² + 1 = 0, D = −4. The roots are +i and −i, with no real x-axis intersections.
Assumptions & practical notes
This calculator requires a nonzero quadratic coefficient, with |a| ≥ 10⁻¹²; a = 0 is a linear equation.
Coefficients have absolute values at most one million. Floating-point calculations are approximate. Near-repeated roots can be sensitive to rounded coefficients.
See our calculation methodology for unit definitions and testing practices.
References & measurement guidance
References explain the underlying mathematics. The examples and interactive calculations on this page are specific to CalcCommons.
Frequently asked questions
What does a negative discriminant mean?
The roots are complex conjugates. The real-valued parabola does not cross the x-axis.
Is this the same as factoring?
Factoring rewrites the expression as a product. This solver also handles irrational and complex roots that the rational factoring mode does not express.
Why are both roots the same?
A zero discriminant gives one repeated root. The curve touches the x-axis at its vertex.
Related calculators
Factoring Calculator
Find integer factors, greatest common factors, and rational factors of quadratic trinomials.
Right Triangle Calculator
Solve a right triangle from two sides or one side and an acute angle.
Scientific Notation Calculator
Convert standard and scientific notation and calculate with positive or negative exponents.