Question

Find all values of $a$ for which the inequality $(a+4)x^2-2ax+2a-6<0$ is satisfied for all $x\in R$.

• A. For all $a\in (-\infty ,-6)$
• B. For all $a\in (-\infty ,6)$
• C. For all $a\in (-\infty ,+\infty )$
• D. For all $a\in (-6,-\infty )$

Answer 1

The correct option is A For all $a\in (-\infty ,-6)$

The inequality $(a+4)x^2-2ax+2a-6<0$ has to be satisfied for all $x\in R$. Only possible when, coefficient of $x^2$ and discriminant value are less than zero.
$\Rightarrow a+4<0$ and $4a^2-4(a+4)(2a-6)<0$
$\Rightarrow a\in (-\infty ,-4)$ and $a^2+2a-24=(a+6)(a-4)>0$
$\Rightarrow a\in (-\infty ,-4)$ and $a\in (-\infty ,-6)\cup (4,\infty )$
$\therefore a\in (-\infty ,-6)$
Hence, option A.