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) $a{x}^{2}+bx+c=0$.

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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 solutiondomain 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).

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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).

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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.
Title  quadratic inequality 
Canonical name  QuadraticInequality 
Date of creation  20130322 15:23:48 
Last modified on  20130322 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 2600 
Classification  msc 12D99 
Related topic  QuadraticFormula 
Related topic  SolvingCertainPolynomialInequalities 
Related topic  TangentOfConicSection 
Related topic  IndexOfInequalities 