A quadratic equation has the standard form ax squared plus bx plus c equals zero, where a cannot be zero. These equations appear throughout algebra, physics, engineering, and finance, describing trajectories, areas, profit functions, and countless other relationships that follow a curved, parabolic path. The quadratic formula gives you both solutions at once: x equals negative b plus or minus the square root of the quantity b squared minus 4ac, all divided by 2a. The quantity under the square root, called the discriminant, carries important information: when it is positive there are two distinct real roots; when it equals zero there is exactly one repeated real root; and when it is negative there are no real roots and the solutions are complex numbers. This calculator solves any quadratic equation instantly. You enter the three coefficients a, b, and c, and it returns both roots (or reports if they are complex), the value of the discriminant, the vertex coordinates of the parabola, the axis of symmetry, and the step-by-step working so you can follow the method. Negative coefficients work as expected; simply enter a negative number. It is useful for students working through algebra or calculus, engineers solving for unknown values, and anyone who needs to find where a parabolic function crosses zero. Enter your coefficients to see all results update as you type.
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x = 3 and x = 2
roots of the equation
Discriminant (Δ)1
Root typeTwo real roots
Vertex(2.5, -0.25)
Axis of symmetryx = 2.5
Working: x = (5 ± √1) / 2. x1 = 3, x2 = 2.
How it works
The calculator computes the discriminant as b squared minus 4ac. When positive, it takes the square root and applies the formula to give two distinct real roots. When zero, both roots are the same repeated value of negative b over 2a. When negative, no real roots exist and the calculator notes that the solutions are complex. The vertex x-coordinate is negative b over 2a; the vertex y-coordinate is found by substituting this back into the original equation. The axis of symmetry is the vertical line through the vertex x-coordinate.
Worked example
With a = 1, b = -5, c = 6 the equation is x squared minus 5x plus 6 equals zero. The discriminant is (-5) squared minus 4(1)(6) = 25 minus 24 = 1. Because the discriminant is positive there are two real roots. Applying the formula: x = (5 plus or minus 1) divided by 2, giving x = 3 and x = 2. The vertex is at x = 2.5, y = 2.5 squared minus 5(2.5) plus 6 = -0.25.
