a ≠ 0
Click any example below to automatically load the coefficients and see real-world applications of quadratic equations.
The discriminant D summarizes how many and what type of roots a quadratic has: D > 0 → two distinct real roots; D = 0 → one repeated real root; D < 0 → two complex conjugate roots. Its magnitude also hints at the distance between roots.
The axis of symmetry is x = -b/(2a). The vertex sits on this line at x_v = -b/(2a) and y_v = f(x_v). While our tool solves for the roots, understanding the vertex helps with graphing and optimization problems.
If a > 0 the parabola opens upward (minimum at the vertex); if a < 0 it opens downward (maximum at the vertex).
Solve 2x^2 - 5x - 3 = 0. Here a = 2, b = -5, c = -3. Discriminant: D = (-5)^2 - 4(2)(-3) = 25 + 24 = 49. Since D > 0, there are two real roots.
For ax^2 + bx + c = 0 with a ≠ 0, the solutions are x = (-b ± √(b^2 - 4ac)) / (2a). The ± gives two branches, producing two roots when the discriminant is positive.
When the discriminant b^2 - 4ac is negative. The roots take the form p ± qi where p and q are real numbers.
Then the equation is linear (bx + c = 0) and not quadratic. Our tool requires a ≠ 0 to use the quadratic formula.
The coefficient a controls opening direction and vertical stretch; b shifts the axis of symmetry; c is the y-intercept (0, c).
Yes. Completing the square derives the quadratic formula and also provides vertex form y = a(x - h)^2 + k for graphing and optimization.