Quadratic Formula Calculator
ax² + bx + c = 0
Root 13.0000
Root 22.0000
Discriminant1.0000
Vertex(2.5000, -0.2500)
The quadratic formula solves any equation of the form ax² + bx + c = 0. This calculator finds the roots, the discriminant that tells you how many real solutions exist, and the coordinates of the parabola vertex. When the discriminant is negative it returns the two complex roots instead of real ones.
Formula
x = (-b ± √(b² - 4ac)) / (2a)
- a
- Coefficient of x² (must be non-zero)
- b
- Coefficient of x
- c
- Constant term
How it works
- Enter the coefficients a, b, and c. The value of a must not be zero, otherwise the equation is linear rather than quadratic.
- The tool computes the discriminant b² - 4ac. A positive discriminant gives two distinct real roots, zero gives one repeated real root, and a negative value gives a pair of complex conjugate roots. It also reports the vertex at x = -b / (2a).
Worked example
Solve x² - 5x + 6 = 0 (a = 1, b = -5, c = 6).
- Discriminant = (-5)² - 4(1)(6) = 25 - 24 = 1.
- Roots = (5 ± √1) / 2 = (5 ± 1) / 2, giving 3 and 2.
- Vertex x = -(-5) / (2·1) = 2.5, vertex y = -0.25.
Two real roots, x = 3 and x = 2, with vertex at (2.5, -0.25).
Frequently asked questions
- What does the discriminant tell me?
- The discriminant b² - 4ac determines the nature of the roots. Positive means two distinct real roots, zero means one repeated real root, and negative means two complex conjugate roots.
- Why must a be non-zero?
- If a equals zero there is no x² term and the equation becomes linear, so the quadratic formula no longer applies. This calculator returns no result when a is 0.
- What happens when the roots are complex?
- When the discriminant is negative the calculator reports the two roots in the form p + qi and p - qi, where the real part is -b / (2a) and the imaginary part comes from the square root of the negative discriminant.