Question

If the roots of the equation ${\lambda }^2+8\lambda +{\mu }^2+6\mu =0$ are real, then $\mu$ lies between

• A. $-2$ and $8$
• B. $-3$ and $6$
• C. $-8$ and $2$
• D. $-6$ and $3$

Answer 1

The correct option is C

$-8$ and $2$

Explanation for the correct option

The given equation: ${\lambda }^2+8\lambda +{\mu }^2+6\mu =0$.

Compare the given equation with the general form of the quadratic equation: $a{x}^2+bx+c=0$.

Thus, $a=1,b=8,c={\mu }^2+6\mu$.

The discriminant of the equation, $D=b^2-4ac$.

$\Rightarrow D={\left(8\right)}^2-4\left(1\right)\left({\mu }^2+6\mu \right)$

$$\begin{gathered} =64-4{\mu }^2-24\mu \\ =-4\left({\mu }^2+6\mu -16\right) \\ =-4\left({\mu }^2+8\mu -2\mu -16\right) \\ =-4\left[\mu \left(\mu +8\right)-2\left(\mu +8\right)\right] \end{gathered}$$

$\Rightarrow D=-4\left(\mu +8\right)\left(\mu -2\right)$

As the roots of the equation are real, thus $D\ge 0$.

$$\begin{gathered} \Rightarrow -4\left(\mu +8\right)\left(\mu -2\right)\ge 0 \\ \Rightarrow \left(\mu +8\right)\left(\mu -2\right)\le 0 \\ \Rightarrow \mu \in \left[-8,2\right] \end{gathered}$$

Therefore, $\mu$ lies between $-8$ and $2$.

Hence, option(C) is the correct option i.e. $-8$ and $2$