Question

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

• A. 0
• B. $\pm\, 1$
• C. $\pm\, 7$
• D. 49

Answer 1

Answer: (C)

$\pm\, 7$

Answer 2

We have the expression
$2x^2 + mxy + 3y^2 - 5y - 2$
Comparing the given expression with
$ax^2 + 2hxy + by^2 + 2gx + 2 fy + c$,
we get
$a= 2, h = \frac{m}{2} , b=3,c=-2 , g=0, f=- \frac{5}{2}$
The given expression is resolvable into linear factors, if
$abc + 2 fgh - af^{2} -bg^{2} -ch^{2} = 0$
$\left(2\right)\left(3\right)\left( -2\right) + 2\left(0\right)=2 \left(\frac{25}{4}\right) - 0 -\left(-2\right) \frac{m^{2}}{4} = 0$
$\Rightarrow - 12 - \frac{25}{2} + \frac{m^{2}}{2} = 0$
$\Rightarrow \frac{m^{2}}{2} = \frac{49}{2}$
$\Rightarrow m^{2} = 49$
$\Rightarrow m = \pm7$