Find the set of values for k?

Question

Find the set of values for k for which the line y=2x−k meets the curve y=x2+kx−2 at two distinct points.

 

I only know you have to use a discriminant. :/

Answer 1

This happens if

 

x² + kx - 2 = = 2x - k

 

has two distinct solutions.

 

x² + (k-2)x - (2 - k) = 0

 

Two distinct solutions is the same as having a positive discriminant.

 

b² - 4ac > 0

 

(k - 2)² + 4(1)(2 - k) > 0

 

(k - 2)² - 4(k - 2) > 0

 

(k - 2)[(k - 2) - 4] > 0

 

(k - 2)(k - 6) > 0

 

k < 2 or k > 6