Hi Sonia!

Once you obtain the equation of the tangent from (i), you will need to solve this equation simultaneously with the cubic curve. You will eventually obtain a cubic polynomial in x which equates to zero. We must then solve the cubic equation accordingly.

Of course we already know that x = 1 is a solution to the curve (from the data in part (i)) so (x - 1) must be a factor of that cubic polynomial.

You can then factorise the polynomial by either long division, comparing coefficients, guessing of values or some other method i.e. the techniques which you have learnt to solve cubic expressions.

Hopefully my working helps!

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As a side note, which is not essential to the question, that tangent line only touches the curve at (1, 6), but does not really cut the curve. So I am actually already expecting x = 1 to be a repeated root (you do not need to know this though; the idea is something similar to the discriminant of roots for real and repeated roots). As it turns out, upon factorisation, indeed the (x - 1) is repeated.