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Quadratic Formula Calculator
MATH CALCULATOR
Solve ax² + bx + c = 0 and find the discriminant, roots, axis of symmetry, and vertex.
Discriminant25
Root 14
Root 2−1
Axis of symmetryx = 1.5
Vertex(1.5, −6.25)
How it works
Enter the three coefficients. The discriminant determines whether roots are two real values, one repeated real value, or a complex-conjugate pair.
Formula
x = [−b ± √(b² − 4ac)] ÷ 2a. The discriminant is b² − 4ac, and the symmetry axis is x = −b ÷ 2a.
Worked example
For x² − 3x − 4 = 0, the discriminant is 25, the roots are 4 and −1, and the vertex is (1.5, −6.25).
Important limits
- Coefficient a cannot be zero; that would be a linear equation.
- Decimal inputs can introduce small floating-point rounding.
- Complex roots are displayed in a + bi form.
Frequently asked questions
What does a negative discriminant mean?
The equation has no real roots and instead has two complex-conjugate roots.
What is a repeated root?
When the discriminant equals zero, both formula branches give the same root.
Sources and review notes
The calculator applies the standard quadratic formula and vertex relationships from elementary algebra. Last reviewed September 29, 2026.