quadratic inequality

The of a quadratic inequality is

ax2+bx+c< 0 (1)


ax2+bx+c> 0 (2)

where a, b and c are known real numbers and  a0.

Solving such an inequalityMathworldPlanetmath, i.e. determining all real values of x which satisfy it, is based on the fact that the graph of the quadratic polynomial function  xax2+bx+c  is the parabola


opening upwards if  a>0  and downwards if  a<0.

For obtaining the solution we first have to determine the real zeroes of the polynomialMathworldPlanetmathPlanetmathPlanetmath ax2+bx+c, i.e. solve the quadratic equation (http://planetmath.org/QuadraticFormula)  ax2+bx+c=0.

  • If there is two distinct real zeroes x1 and x2 (say  x1<x2),  then the parabola intersects the x-axis in these points.  In the case  a>0  the parabola opens upwards and thus  y<0  in the interval(x1,x2), but  y>0  outside this interval.  I.e., for positive a, the solution of (1) is


    and the solution of (2) is

    x<x1 or x>x2

    (note that the latter solution-domain consists of two distinct portions of the x-axis and therefore must be expressed with two separate inequalities, not with a double inequality as the former).  For negative a we must swap those solutions for (1) and (2).

    Figure 1: Solving for ax2+bx+c<0 when a>0 and the quadratic has two distinct roots
  • If there is only one real zero of the polynomial (we may say that  x2=x1), the parabola has x-axis as the tangentPlanetmathPlanetmath (http://planetmath.org/TangentLine) in its apex.  For positive a the other points of parabola are above the x-axis, i.e. they have  y>0  always  but  y<0  never.  So, (1) has no solutions, but (2) is true for all  xx1 (i.e.  x<x1  or  x>x1).  For the case of negative a we again must change those solutions for (1) and (2).

    Figure 2: Solving for ax2+bx+c>0 when a>0 and the quadratic has only one root
  • There can still appear the possibility that the polynomial has no real zeroes (the roots of the equation are imaginary).  Now the parabola does not intersect or touch the x-axis, but is totally above the axis when a is positive (y>0  always) and totally below the axis when a is negative (y<0  always).  Thus we get no solutions at all (the inequality is impossible) or all real numbers x as solutions, according to what the inequality (1) or (2) demands.

    Figure 3: ax2+bx+c>0 for all x when a>0 and the quadratic has no roots
Title quadratic inequality
Canonical name QuadraticInequality
Date of creation 2013-03-22 15:23:48
Last modified on 2013-03-22 15:23:48
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 12
Author pahio (2872)
Entry type Topic
Classification msc 97D40
Classification msc 26-00
Classification msc 12D99
Related topic QuadraticFormula
Related topic SolvingCertainPolynomialInequalities
Related topic TangentOfConicSection
Related topic IndexOfInequalities