Calculadora de Desigualdades Quadráticas
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
ax² + bx + c > 0 ⇔ x < r_min ou x > r_max (quando a > 0)
About this calculator
This calculator solves quadratic inequalities of the form ax² + bx + c > 0, where a, b, and c are real numbers and a is nonzero. It finds the roots of the associated quadratic equation (ax² + bx + c = 0) and, based on the sign of the leading coefficient a, determines the solution set of the inequality. The result is presented as intervals on the real line indicating where the expression is positive.
The operation is simple: the calculator applies the quadratic formula to find the real roots (if any). If a > 0, the parabola opens upward, and the expression is positive outside the interval between the roots. If a < 0, the parabola opens downward, and the expression is positive between the roots. If the discriminant is negative, the inequality is always true (if a > 0) or never true (if a < 0).
Use this tool in high school math problems, college entrance exam preparation, or in engineering and economics contexts involving sign analysis of quadratic functions. For example, to determine temperature ranges where a chemical process is feasible, or to solve inequalities in optimization problems.
Caution: the calculator assumes a is nonzero. If a = 0, the inequality becomes linear, and this tool is not suitable. Also, remember that the solution changes depending on the sign of a; check that you entered the correct coefficient. Always verify that the result makes sense in the context of your problem.
Frequently asked questions
What does the inequality ax² + bx + c > 0 mean?
It means finding all x values for which the quadratic expression is positive, i.e., the parabola graph is above the x-axis.
What if the discriminant is negative?
If the discriminant is negative, the equation has no real roots. In that case, if a > 0, the inequality is true for all real x; if a < 0, it is never true.
Can I use this calculator for inequalities with ≥ or < ?
Not directly. This calculator solves only > 0. For ≥, include the roots; for < or ≤, use the complement or other tools.
What happens if a is negative?
If a < 0, the parabola opens downward. The solution of ax² + bx + c > 0 will be the interval between the roots (if they exist).
How does the calculator handle complex roots?
It indicates that there are no real roots and gives the solution based on the sign of a, as described in the question about negative discriminant.