Two real roots
x²−5x+6=0 has roots 2 and 3, D=1, and vertex (2.5, −0.25).
Solve ax²+bx+c=0, including real, repeated, or complex roots, and inspect the discriminant, vertex, symmetry, factorization, and parabola graph.
A quadratic equation has degree two and standard form ax²+bx+c=0, where a cannot be zero.
Its graph is a parabola. The coefficient a controls whether it opens upward or downward.
This formula finds all real or complex roots.
x = (-b ± √(b²−4ac))/(2a)Positive D gives two real roots, zero gives a repeated root, and negative D gives complex conjugates.
D = b²−4acThe axis of symmetry is x=h.
h = -b/(2a); k = f(h)x²−5x+6=0 has roots 2 and 3, D=1, and vertex (2.5, −0.25).
x²−6x+9=0 has the repeated root 3.
x²+1=0 has roots i and −i.
For real roots r₁ and r₂, the factored form is a(x−r₁)(x−r₂).
The calculator uses a cancellation-resistant root method internally while showing the familiar quadratic formula.
A negative discriminant produces conjugate roots with the same real part and opposite imaginary parts.
The graph range is centered on the vertex and any real roots, with bounded expansion for extreme equations. The y-intercept is (0,c).
It is a degree-two equation ax²+bx+c=0 with a nonzero.
x=(-b±√(b²−4ac))/(2a).
It is b²−4ac and determines the type and number of roots.
The equation has two complex conjugate roots and no real x-intercepts.
When D=0, both formula branches produce the same root.
Use h=−b/(2a), then evaluate the quadratic at h to find k.
Every quadratic factors over complex numbers, but a clean real factored form is shown only for real roots.
The equation is not quadratic, so this calculator rejects it.