Determine the equations of the lines that have a slope of 2 and that intersect the quadratic function f(x)= x(x-6) once, twice and never

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The equation of the lines has to be determined that have a slope 2 and intersect y = x(x - 6) once, twice and never.

The general equation of a line with slope 2 is y = 2x + k.

If the line intersects the given curve:

2x + k = x(x - 6)

=> x^2 - 6x - 2x - k = 0

=> x^2 - 8x - k = 0

This has one solution when (-8)^2 = 4*(-k)*1

=> 64 = -4k

=> k = -16

It has no real solutions when 64 < -4k

=> 64 > 4k

=> -16 > k

It has two solutions for all values of k > -16

The lines y = 2x + k intersect y = x(x - 6) once if k = -16, twice if k > -16. There is no intersection if k < -16 

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