Determine the positive values of `' k '` for which the equation `x^2+k x+64=0` and `x^2-8x+k=0` will both have real roots.

Explanation

Given equations are
`x^(2)+kx+64=0" ".....(1)`
and `x^(2)-8x+k=0" ".....(2)`
Let `D_(1)andD_(2)` be discriminants of equations (1) and (2) respectively, then
`D_(1)=k^(2)-4xx64orD_(1)=k^(2)-656`
and `D_(2)=(-8)^(2)-4korD_(2)=64-4k`
Both equations will have real roots, if
`D_(1)gt=0" "and" "D_(2)gt=0`
`impliesk^(2)-256gt=0" "and" "64-4kgt=0`
`impliesk^(2)gt=256" "and" "64-4kgt=0`
`implieskgt=16" "and" "kglt=16`
`impliesk=16`
Hence, both equations will have real roots when k=16.