TSR Wiki > Study Help > Exams and Qualifications > A Levels > STEP > STEP I 2007 question 6 solution
(i)
We are given 
and also 
You can change the 
into 
(difference of two squares)
So we now have 
and we can substitute in the expression for d
We also know that 
and also 
which we can substitute into the above equation.
Substituting Y
Substituting X
We are now told that 
and also 
Let 
and 
and also change the range: 
We now must try different values for 
in order to get values for x and y that satisfy our range.
and 
Considering that 
Either 
or 
but we know already that 
Therefore . Conveniently ;) 
gives us the nice values of 
Ergo 
and 
(ii)
As before, we realize that we must reduce the expression on the right. We used the difference of two squares in the previous part and so this time we shall use the different of two cubes.
We now have: 
Which is equivalent to 
You should, at this point, realize that the left hand side of the above equation should factorize into something with a 
.
We notice this by simply looking at what our end product should be (
).
The expansion of is 
and so we can alter our own equation accordingly:
Which when rearranged gives
We have also been told that which we can rearrange to and then substitute into the equation we have just worked out.
The above equation, you should notice is now a quadratic equation and we can use the quadratic formula:
Note that following the same method as above we get 
Trial and error works at this point but the problem can be simplified by considering the algebra within the square root, which must be an integer. Therefore:
where n and d are integers.
We then try values 1,2,3... for n.
yields 
but 
and 
are positive integers
gives a non integer.
gives 
Substituting this into gives a balanced equation hence 
are the values required
Solution by The Muon
