Question

Find the largest integral value of $m$ for which the quadratic expression
$y=x^2+(2m+4)x+7m+8$ is positive for all $x\in R$

Answer 1

Answer: 3

Given expression $>0$ for all $x$.
$\iff \Delta <0$ as leading coefficient $>0$
$\iff 4(m+2{)}^2-4(7m+8)<0$
$\iff m^2+4m+4-7m-8<0$
$\iff m^2-3m-4<0$
$\iff (m-4)(m+1)<0$
$\iff -1<m<4$
$\Rightarrow$ Largest integral value of $m=3$