Question

for what value(s) of k does the equation kx - 10 - 5x^2 have:
a) one real solution
b) two distinct real solutions
c) no real solution

Answer 1

I will assume your "equation" is equal to zero, that is,
kx - 10 - 5x^2 = 0
or
5x^2 - kx + 10 =0

your discriminant is b^2 - 4ax, or
k^2 - 4(5)(10)
= k^2 - 200

to have one real root:
k^2 - 200 = 0
k^2 = 200
k = ±√200 or ±10√2

for 2 distinct real roots:
k^2 - 200 > 0
k^2 > ± 10√2
k < -10√2 OR k > 10√2

for no reals:
k^2 < 0
-10√2 < x < 10√2 , that is, x is between ±10√2