Quadratic Equation Solver

Solve a x²+b x+c=0 with the quadratic formula. The discriminant detects real, double, and complex roots, and the vertex, axis, and parabola graph are shown.

Input

Enter the coefficients a, b and c of the quadratic equation a x²+b x+c=0. The two roots are found with the quadratic formula.

Coefficient a

Coefficient b

Coefficient c

The equation is a x²+b x+c=0. The coefficient a must not be zero.

Result

Two distinct real roots

Root x₁

Root x₂

Discriminant D

Axis of symmetry

x = 1.5

Vertex

( 1.5 , -0.25 )

The roots are x = ( -b ± sqrt(D) ) / 2a with discriminant D = b² - 4ac. When D is negative the roots are complex.

How it works

The quadratic formula x = ( -b ± sqrt(D) ) / 2a is used, with discriminant D = b² - 4ac classifying the roots.
When D is positive there are two distinct real roots, when D is zero there is a double root, and when D is negative there are conjugate complex roots.
The parabola vertex is ( -b / 2a , c - b² / 4a ) and the axis of symmetry is x = -b / 2a.
When the coefficient a is zero it is not a quadratic equation, so it cannot be solved.

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