Question

The value(s) of 'p' for which the parabola represented by quadratic function $y=(p-2)x^2+8x+(p+4)$ will remain below X-axis for all real values of x is .

• A. $p\in (-\infty ,-6)\cup (4,\infty )$
• B. $p\in (-\infty ,-6)$
• C. $p\in (4,\infty )$
• D. $p\in (-6,4)$

Answer 1

The correct option is B $p\in (-\infty ,-6)$

For a parabola to remain below X-axis, it means the parabola will be downward facing and will not intersect/touch X-axis.
$\Rightarrow a<o\Rightarrow p-2<0\Rightarrow p<2...\left(i\right)andD<0\Rightarrow (8{)}^2-4(p-2\left)\right(p+4)<0\Rightarrow 16-p^2-4p+2p+8<0\Rightarrow -p^2-2p+24<0\Rightarrow p^2+2p-24>0\Rightarrow (p+6\left)\right(p-4)>0...(ii\left)From\right(i\left)and\right(ii),weget-p<-6$