Question

Determine the number of positive values of '$a$' for which the equations $x^2+ax+64=0$ and $x^2-8x+a=0$ will both have real roots.

Answer 1

Equation 1 : - $x^2+ax+64=0$

for real solution,

discriminant $\left(\Delta \right)$ should be $\ge 0$

$\Rightarrow a^2\ge -4\times 64\times 1\ge 0$

$\Rightarrow a^2\ge 256$

$\Rightarrow |a|\ge 16\Rightarrow aϵ(-\infty ,-16]\cup [16,\infty )$

Equation 2 : - $x^2-8x+a=0$

$\therefore$ for real solution.

$(-8{)}^2-4\times a\times 1\ge 0$

$\Rightarrow 64-4a\ge 0$

$\Rightarrow 4a\le 64$

$\Rightarrow a\le 16\Rightarrow aϵ(-\infty ,16]$

$\therefore$ No of positive values of ${}^{\prime }{a}^{\prime }=1$