They intersect if there's some x where kx = sqrt(x) - 1
To solve that, get the square root alone on one side.
kx + 1 = sqrt(x)
Square both sides
k^2x^2 + 2kx + 1 = x
k^2x^2 + (2k-1)x + 1 = 0
Find the discriminant b^2 - 4ac. It has a solution if b^2 - 4ac >= 0. That gives you your condition on k.
To solve that, get the square root alone on one side.
kx + 1 = sqrt(x)
Square both sides
k^2x^2 + 2kx + 1 = x
k^2x^2 + (2k-1)x + 1 = 0
Find the discriminant b^2 - 4ac. It has a solution if b^2 - 4ac >= 0. That gives you your condition on k.
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(x^(1/2) - 1 = kx ---> x^(1/2) = 1+kx , square both sides ---> x = k^2x^2 + 2kx + 1
or k^2x^2 +(2k-1)x + 1 = 0
discriminant = (2k-1)^2 - 4k^2 = 0 ---> k = 1/4
or k^2x^2 +(2k-1)x + 1 = 0
discriminant = (2k-1)^2 - 4k^2 = 0 ---> k = 1/4