Quadratic Expressions-II

Quadratic Expressions-II
Quadratic Inequation :
The inequations of the form ax²+bx+c>0, ax²+bx+cax²+bx+c<0 and ax²+bx+c \(\leqslant 0\), where are called quadratic inequations
 For example : (i) x²-5x+6<0                (ii) \({x^2} + 7x + 8 \geqslant 0\)
NOte : Every quadratic inequation have infinite number of solutions.
Domain and Range of a quadratic inequation:
    Consider a quadratic inequationx²-4x+3<0
    Now, x²-4x+3<0
    (x-1)(x-3)<0
    x-1>0 and x-3<0
         (OR)
    x-1<0 and x-3>0
    x>1 and x<3
        (OR)
    x<1 and x>3
    Represent the above inequalition on a number line
Case (i) x>1 and x <3

Case (ii)
    here no solution exists i.e \(\phi \)
    The combined solution set is
    =\(\left\{ {x/1 < x < 3} \right\} \cup \phi \)
    =\(\left\{ {x/1 < x < 3,x \in R} \right\}\)
    Hence, we will get the following cases.
    Let f(x)=ax²+bx+c
    where  \(f(x) = \left( {x - \alpha } \right)\,\left( {x - \beta } \right)\)
Case(i) : ax²+bx+c <0
          \(\Rightarrow \left( {x - \alpha } \right)\,\left( {x - \beta } \right) < 0\)
    The solution set =\(\left\{ {x/\alpha < x < \beta ,x \in R} \right\}\)
Case(ii) : ax²+bx+c \(\le\)0 
\(\Rightarrow \left( {x - \alpha } \right)\,\left( {x - \beta } \right) \leqslant 0\)
The solution set =\(\left\{ {x/\alpha \leqslant x \leqslant \beta ,\,x \in R} \right\}\)
Case (iii) ax²+bx+ c >0
        \(\Rightarrow \left( {x - \alpha } \right)\,\left( {x - \beta } \right) > 0\)
    The solution set =   \(\left\{ {x3x < \alpha \,\,or\,x > \beta ,x \in R} \right\}\)   
Case (iv): \(a{x^2} + bx + c \geqslant 0\)
    \(\Rightarrow \left( {x - \alpha } \right)\,\left( {x - \beta } \right) \geqslant 0\)
    The solution set =\(\left\{ {x/x \leqslant \alpha \,\,or\,\,x \geqslant \beta ,\,\,x \in R} \right\}\)
Note : Range will be calculated depending upon the domain of the quadratic function.