Question

The values of $m$ for which the expression $2x^2+mxy+3y^2-5y-2$ can be expressed as the product of two linear factors are

• A. $+7,-7$
• B. $+5,-5$
• C. $+4,-4$
• D. $+1,-1$

Answer 1

The correct option is A $+7,-7$

Given: expression $2x^2+mxy+3y^2-5y-2$

To find the value of $m$ for which the given expression is a product of two linear factors

Sol: Let us consider the given expression as quadratic equation variable $y$, i.e., $3y^2+(mx-5)y+(2x^2-2)$. Then, $a=3,b=mx-5,c=2x^2-2$

And we know a quadratic equation can be factored into linear factors if the discriminant of the equation is a perfect square.

i.e., $b^2-4ac=(mx-5{)}^2-4(3\left)\right(2x^2-2)⟹(mx{)}^2+25-10mx-24x^2+24⟹(m^2-24)x^2-10mx+49⟹(m^2-24)x^2-2\left(7\right)\left(\frac{5mx}{7}\right)+7^2$

Therefore $b^2-4ac$ is a perfect square trinomial precisely when

$(m^2-24)x^2={\left(\frac{5mx}{7}\right)}^2$ for all $x$

This is equivalent to $m^2-24=\left(\frac{25}{49}\right)m^2⟹49m^2-49\times 24=25m^2$ or equivalently $m^2=49⟹m=\pm 7$