quadratic inequality

The of a quadratic inequality is

(1)

or

$a{x}^2+bx+c>\mathrm{\hspace{0.33em}0}$ (2)

where $a$, $b$ and $c$ are known real numbers and  $a\ne 0$.

Solving such an inequality, i.e. determining all real values of $x$ which satisfy it, is based on the fact that the graph of the quadratic polynomial function  $x\mapsto a{x}^2+bx+c$  is the parabola

$$y=a{x}^2+bx+c,$$

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

For obtaining the solution we first have to determine the real zeroes of the polynomial $a{x}^2+bx+c$, i.e. solve the quadratic equation (http://planetmath.org/QuadraticFormula)  $a{x}^2+bx+c=0$.

• If there is two distinct real zeroes $x_1$ and $x_2$ (say  ),  then the parabola intersects the $x$-axis in these points.  In the case  $a>0$  the parabola opens upwards and thus    in the interval  $(x_1,x_2)$, but  $y>0$  outside this interval.  I.e., for positive $a$, the solution of (1) is

  and the solution of (2) is

  (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 when $a>0$ and the quadratic has two distinct roots

• If there is only one real zero of the polynomial (we may say that  $x_2=x_1$), the parabola has $x$-axis as the tangent (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    never.  So, (1) has no solutions, but (2) is true for all  $x\ne x_1$ (i.e.    or  $x>x_1$).  For the case of negative $a$ we again must change those solutions for (1) and (2).

  

  Figure 2: Solving for $a{x}^2+bx+c>0$ 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 (  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: $a{x}^2+bx+c>0$ 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