Alan

Expand A(y - B)(y - C)

A*(y² - (B+C)y + BC)

Ay² - A(B+C)y + ABC

Compare the coefficients with those of  6y² - y - 15

Can you take it from here?

Don't forget  -8 and  \(\pm 64^{1/4}\) (fourth root of 64 is real, but not an integer). 

Also 2 is in the domain of the function.

Multiply all terms by \(x^{2/3}\)  (and remember that the cube root of x is the same as x^1/3).

Make use of the fact that \(x^a\times x^b=x^{a+b} \)

Collect like terms together.

Solve the resulting equation for x.

For (a) simply substitute the values x = 5 and y = 12 into both expressions and calculate the results.

For (b)  set (x+y)² = x² + y² then expand the left hand side and simplify the resulting equation.  You will then have a condition that must be satisfied for the two ex[ressions to be equal.

The following should help you decide:

I used n as a general value. You can replace it with n1 and the first two terms produce a real number; replace it with n2 and the second two terms produce a real value.  So your original expression is true for all positive integer values of n1 and n2.  

The question asks for a proof "if and only if"  (iff).  The expression is a real number for all the possibilities listed, however, it is also true for other possibilities not listed (such as n1 = n2+3, for example), which means none of a) to d) provide the correct answer (they all conform to the "if" part, but not the "only if" part).

No, I didn't assume n1 = n2.  The conclusion holds for each pair of terms separately.  

This means 2*k - 3 = -21

Can you now rearrange this to find k?

As follows:

The quoted answer d) is rather strange, since the question specifies that the n's are positive integers!. 

What you have done is to find the vertical height of the platform from ground to top.  Now you have that and the 110' you can use Pythagoras' theorem to find the length of the zip line.