For what value of k do the two graphs with equations y=sqrt(x)-1 and y=kx intersect
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For what value of k do the two graphs with equations y=sqrt(x)-1 and y=kx intersect

[From: ] [author: ] [Date: 12-04-01] [Hit: ]
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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.

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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
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