Quadratic Inequality Solver

Enter a x squared plus b x plus c and an inequality sign to find the solution range. It uses the discriminant and the parabola direction, shown with a graph.

Input

Enter coefficients a, b, c and an inequality sign to find the solution range of the quadratic inequality.

a (x squared term)

b (x term)

c (constant)

Inequality

Inequality to solve: 1 x squared + -1 x + -6 > 0

Result

Solution range

x less than -2, or x greater than 3

Discriminant D

Sign of discriminant

Positive (two distinct real roots)

Smaller root

-2

Larger root

Vertex

( 0.5 , -6.25 )

The sign of the discriminant D equals b squared minus 4 a c determines the intersections, and the parabola direction with the inequality direction sets the interval where the quadratic is positive or negative as the solution.

How it works

Enter coefficients a, b, c and an inequality sign to solve the quadratic inequality a x squared plus b x plus c against zero. The value of a must not be zero, otherwise it is not a quadratic inequality.
First the discriminant D equals b squared minus 4 a c is computed. A positive D gives two distinct real roots, a zero D gives a repeated root (the parabola is tangent), and a negative D gives no real roots.
The direction of the parabola (the sign of a) is combined with the chosen inequality direction to decide where the quadratic function is positive or negative, which sets the solution range.
When D is positive, the solution is either between the two roots or outside them. When the sign allows equality, the endpoints are included in the solution.
When D is negative, the parabola does not cross the x axis, so the solution is either all real numbers or no solution.
The parabola graph marks the solution range on the x axis with a thick orange line and shows the real roots and the vertex.

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