Question

Find the greatest value of a non-negative real number $\lambda$ for which both the equations $2x^2+(\lambda -1)x+8=0$ and $x^2-8x+\lambda +4=0$ have real roots.

• A. $\lambda =12$
• B. $\lambda =16$
• C. $\lambda =15$
• D. $\lambda =14$

Answer 1

Answer: (A)

$\lambda =12$

$2x^2+(\lambda -1)x+8=0$ ….$\left(1\right)$

$x^2-8x+\lambda +4=0$ ….$\left(2\right)$

Both equations have real roots

So, $\Delta \ge 0$ for both equations

For equation $\left(1\right)$

$\Delta =(\lambda -1{)}^2-64\ge 0$

$(\lambda -1{)}^2\ge 64$

$\lambda \in (-\infty ,-7]\cup [9,\infty )$ ….$\left(3\right)$

For equation $\left(2\right)$

$\Delta =64-(\lambda +4)\times 4\ge 0$

$\lambda \le 12$ …..$\left(4\right)$

From $\left(3\right)$, $\left(4\right)$

$\lambda \in (-\infty ,7]\cup [9,12]$

Greatest non-negative value of $\lambda =12$.