Question

The value of $m$ for which the expression $2x^2+mxy+3y^2-5y-2$, can be split into two factors, is

• A. $\pm 7$
• B. $\pm 2$
• C. $0$
• D. $\pm 5$

Answer 1

The correct option is A $\pm 7$

Condition for quadratic expression
$a{x}^2+2hxy+b{y}^2+2gx+2fy+c$ has two linear factors
$\left|\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right|=0$
Solving the above determinant, we get

$\Rightarrow abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0$

Here, $a=2,h=\frac{m}{2},b=3,g=0,f=\frac{-5}{2},c=-2$

$\therefore \left(2\right)\left(3\right)(-2)+[2\times \left(\frac{-5}{2}\right)(0\left)\left(\frac{m}{2}\right)\right]-\left(2\right){\left(\frac{-5}{2}\right)}^2-\left(3\right)(0{)}^2-(-2){\left(\frac{m}{2}\right)}^2=0$

$\Rightarrow -12-\frac{25}{2}+\frac{m^2}{2}=0$

$\Rightarrow m^2=49$

$\Rightarrow m=\pm 7$

Hence, option $A$ is the correct answer.