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quadratic inequality
The normal form of a quadratic inequality is
$\displaystyle ax^{2}\!+\!bx\!+\!c\;<\;0$  (1) 
or
$\displaystyle ax^{2}\!+\!bx\!+\!c\;>\;0$  (2) 
where $a$, $b$ and $c$ are known real numbers and $a\neq 0$.
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 $x\mapsto ax^{2}\!+\!bx\!+\!c$ is the parabola
$y\;=\;ax^{2}\!+\!bx\!+\!c,$ 
opening upwards if $a>0$ and downwards if $a<0$.
For obtaining the solution we first have to determine the real zeroes of the polynomial $ax^{2}\!+\!bx\!+\!c$, i.e. solve the quadratic equation $ax^{2}\!+\!bx\!+\!c=0$.

If there is two distinct real zeroes $x_{1}$ and $x_{2}$ (say $x_{1}<x_{2}$), 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 $(x_{1},\,x_{2})$, but $y>0$ outside this interval. I.e., for positive $a$, the solution of (1) is
$x_{1}\;<\;x\;<\;x_{2}$ and the solution of (2) is
$x\;<\;x_{1}\,\,\,\mbox{ or }\,\,\,x\;>\;x_{2}$ (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).

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 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 $x\neq x_{1}$ (i.e. $x<x_{1}$ or $x>x_{1}$). For the case of negative $a$ we again must change those solutions for (1) and (2).

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.
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