Find the equation of the tangent at the curve 2y= 3x^2 -2x-8 cuts the X axis and show that they make supplementary angles with the x axis

let y = 0 since tangent cuts X axis.
Solving it, we get x = -4/3 and 2
y' = 3x-1
hence, slopes are -5 and 5.

therefore, eqn are:
3y + 15x = 4 and 5x-y = 10

The angles are tan-15 and tan-1-5
tan-15 + tan_1-5 = tan-10 =  0 or pi
But  tan-15 < tan-1 infinity (i. e pi/2)
Therefore tan-1-5 is negative of an acute angle. 
Since  an acute angle and its negative cannot complete a full circle (i.e 0), they must be equal to pi.
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Here is another and simpler reason why they must add up to 0 and not pi: From the equations of tangents, since coefficients of x and y are not equal, they are not parallel. hence angle between them cant be 0. hence it must be pi.
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