Question

The positive value of $k$ for which the equations $x^2+kx+64=0$ and $x^2-8x+k=0$ will both have real roots is ________.

• A. $4$
• B. $8$
• C. $12$
• D. $16$

Answer 1

The correct option is D $16$

For a quadratic equation to have real roots, discriminant must be greater than or equal to zero.

For the first equation,

$k^2-4\left(1\right)\left(64\right)\ge 0$ ($\because discriminant=b^2-4ac$)

$\Rightarrow$ $k^2-256\ge 0$

$\Rightarrow$ $(k-16)(k+16)\ge 0$

$\Rightarrow$ $k\ge 16$ and $k\le -16$

For the second equation,

$64-4k\ge 0$

$\Rightarrow$ $k\le 16$

$\therefore$ the value of $k$ that satisfies both the conditions is $k=16$.

$\therefore$Option D is correct.