Question

For what positive values of $m$ can the expression $2x^2+mxy+3y^2-5y-2$ be expressed as the product of two linear factors-

Answer 1

Given equation $2x^2+mxy+3y^2-5y-2=0$
$x=\frac{-my\pm \sqrt{m^2{y}^2-8(3y^2-5y-2)}}{4}$
$x=\frac{-m\pm \sqrt{{(m}^2-24)y^2+40y+16}}{4}$
$x=\frac{-m\pm \sqrt{{(m}^2-24)y^2+2\times 5y\times 4+4^2}}{4}$
To make the square root quantity a perfect square
So, here $m^2-24=5^2$
$\Rightarrow m=\pm 7$
Positive value of $m$ is $7.$