Math
Quadratic Formula Calculator
Solve ax^2 + bx + c = 0. Real or complex roots, discriminant, vertex, and axis of symmetry.
ax^2 + bx + c = 0
Two real roots
- x = 3
- x = 2
Discriminant (b^2 - 4ac)
1
Vertex (x, y)
(2.5, -0.25)
Axis of symmetry
x = 2.5
Frequently Asked Questions about the Quadratic Formula Calculator
What is the quadratic formula?
For any equation in the form ax^2 + bx + c = 0, the solutions are x = (-b +/- sqrt(b^2 - 4ac)) / (2a). Enter your values for a, b, and c and the calculator applies the formula instantly, showing both roots and the discriminant.
What is the discriminant?
The discriminant is b^2 - 4ac. When it is positive, the equation has two distinct real roots. When it equals zero, there is exactly one real root (a repeated root). When it is negative, the equation has two complex roots with no real solutions.
Why does a have to be nonzero?
If a = 0, the x^2 term disappears and the equation becomes bx + c = 0, which is linear, not quadratic. Solve that with x = -c/b. The calculator returns no result when a = 0 because the quadratic formula requires division by 2a.
What is the vertex of a parabola?
The vertex is the highest or lowest point on the parabola. For ax^2 + bx + c, it sits at x = -b/(2a). The calculator substitutes that x back into the equation to get the y-coordinate, and shows the vertex as an (x, y) pair.
When does a quadratic have no real solution?
When the discriminant b^2 - 4ac is negative, the parabola sits entirely above or below the x-axis and never crosses it. The calculator still gives you both complex roots in the form p + qi and p - qi, where p = -b/(2a) and q = sqrt(-(b^2 - 4ac)) / (2a).