Mathematics - Quadratic Equation Question with Solution | TestHub

As given a and bare the roots of the equat $x^2+ax+b=0$ $\Rightarrow$ sum of roots, $a+b=-a$ $\Rightarrow \mathrm{b}=-2\mathrm{a}$ and product of roots, $ab=b$ $$\begin{array}{l}\Rightarrow ab-b=0\\ \Rightarrow b(a-1)=0\end{array}$$ if $b=0$ then $a=0$ if $b\ne 0$ then $a=1$ and $b=-2$ so, the expression will be, $f(x)=x^2+x-2$ $$=x^2+2\cdot \frac{1}{2}x+{\left(\frac{1}{2}\right)}^2-{\left(\frac{1}{2}\right)}^2-2$$ $\Rightarrow \mathrm{f}(\mathrm{x})={\left(\mathrm{x}+\frac{1}{2}\right)}^2-\frac{9}{4}$ So, $\mathrm{f}(\mathrm{x})$ will be minimum, if ${\left(\mathrm{x}+\frac{1}{2}\right)}^2=0$ i.e. when $x=-\frac{1}{2}$ $\Rightarrow$ minimum value of function $=-\frac{9}{4}$