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Quadratic Equation Solver
Enter a, b and c for ax² + bx + c = 0 to see the discriminant, the roots, and the vertex of the parabola.
The quadratic formula, step by step
For ax²+bx+c=0 (with a≠0), first compute the discriminant Δ=b²−4ac. This number determines what kind of roots the equation has. If Δ is positive, the formula x=(−b±√Δ)/2a gives two distinct real solutions. If Δ is exactly zero, both solutions are equal to −b/2a. And if Δ is negative, its square root isn't a real number, so the roots are written as complex numbers p±qi, where p=−b/2a and q=√|Δ|/2a.
A worked example
For x²−3x+2=0 we have a=1, b=−3 and c=2, so Δ=(−3)²−4×1×2=9−8=1. Since Δ is positive, there are two real roots: x=(3±1)/2, giving x₁=2 and x₂=1. The vertex of this parabola sits at x=−b/2a=1.5 and y=c−b²/4a=2−2.25=−0.25.
Frequently asked questions
What is the quadratic formula?
For ax² + bx + c = 0 (with a ≠ 0), the discriminant is Δ = b² − 4ac, and the roots are x = (−b ± √Δ) / 2a.
What does the discriminant tell you?
If Δ is positive, there are two distinct real roots. If Δ is exactly zero, there is one repeated real root. If Δ is negative, the equation has no real roots — the two roots are complex conjugates.
What is the vertex of the parabola?
The vertex sits at x = −b/2a, and its y-coordinate is found by substituting that x back into the equation (or using y = c − b²/4a). If a is positive the parabola opens upward and the vertex is a minimum; if a is negative it opens downward and the vertex is a maximum.
Can a be zero?
No — if a = 0 the equation becomes linear (bx + c = 0), not quadratic, and the quadratic formula no longer applies.