# 2011 S2 Q2 method question

In the Hints for the question, the method for finding the upper bound on z^3 in part (ii) is given as using the fact that 1/3(4kz-z^4) is a square number. In the solutions it says that this is found by comparison with part (i), where the lower bound is found by 1/3(4z^3 -k^3).

I don't really understand how we go from the statement in part (i) to the statement in part (ii), it seems like they are, if equivalent in form, just arbitrarily chosen and I don't really get the methodology for deriving it as needed for the question.

Could someone help explain?
Thanks

### Comparisons

When the solutions say "Comparison with part (i)" it really means try and deduce that something is a perfect square.

The difference between parts (i) and (ii) is that in the first one $x+y=k$ and in the second $x+k=z^2$ so in a sense $k$ is being replaced by $z^2$. In the first part you considered $\frac {4z^3-k^2}3$, and if we use $k=z^2$ in this we get $\frac {4 \times z^2 \times z - k^2}3=\frac {4kz-z^4}3$.

Does this help?

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