Question

Find for what real values of x the expressions
(a) $x^2-2x-3$
(b) $2x^2+5x-3$ are positive or negative.
(c) If $x^2+2ax+10-3a>0$, for all $xϵR$, then
(a) $a<-5$
(b) $-5<a<2$
(c) $a>5$
(d) $2<a<5$

Answer 1

(a) $x^2-2x-3=(x-3)(x+1)=[x-(-1\left)\right](x-3)$
$a=-1$ and $b=3,a<b$.
By $\S 5$ the above expression is $+ive$ if x does not lie between a and b i.e. if x does not lie between $-1$ and $3$. Again it will be $-ive$ if x lies between a and b i.e. x lies between $-1$ and $3$.
(b) $+ive$ if x does not lie between $-3$ and $\frac{1}{2}$.
(c) Ans.(b). $x^2+2ax+10-3a=+ive\forall$ real values of x if disc $4a^2-4(10-3a)=-ive$ and sign of the first term i.e., $1$ is $+ive$.
$\therefore$ $a^2+3a-10<0$
or $(a+5)(a-2)<0$ $\therefore$ $-5<a<2$