Question

Find the values of $m$, for which the equation:
$5x^2-4x+2+m(4x^2-2x-1)=0$ has

• A. equal roots
• B. the product of the roots is $2$
• C. the sum of the roots is $6$
• D. None of the above

Answer 1

Answer: (D)

None of the above

$5x^2-4x+2+m(4x^2-2x-1)$

$\Rightarrow (5+4m)x^2-(4+2m)x+(2-m)=0$

Comparing this with standard quadratic equation we have,

$a=5+4m;b=-(4+2m);c=2-m$

$\left(i\right)$ Equal roots

For equal roots we must have

$\Rightarrow b^2-4ac=0$

$\Rightarrow (4+2m{)}^2-4(5+4m\left)\right(2-m)$

On simplifying and rerranging we get,

$\Rightarrow 5m^2+m-6=0$

$\Rightarrow (5m+6)(m-1)=0$

$\Rightarrow m=-\frac{6}{5},1$

$\left(ii\right)$ Product of roots is $2$

Product of roots $=\dfrc{-b}{a}

$\Rightarrow \frac{2-m}{5+4m}=2$

On simplifying we get,

$\Rightarrow 9m=-8$

$\Rightarrow m=\frac{-8}{9}$

$\left(iii\right)$ Sum of roots is $6$

$\Rightarrow \frac{4+2m}{5+4m}=6$

On simplifying we get,

$\Rightarrow 22m=-26$

$\Rightarrow m=\frac{-13}{11}$