The problem is:
I have tried to factor it, but that doesn't seem to work. Multiplying by the conjugate looks like it involves much more work than is necessary. Any suggestions? Help is appreciated.
Let and . Then , which factorises as . We want to find , so let . Then
Also, . Hence , so that , and equation (*) becomes , or . The only real solution to that cubic equation is z=1, which is what we wanted.
Taking this a bit further, the fact that x and y satisfy the equations and imply that , which is the equation for the golden ratio . Thus , as you can verify by checking that .
That particular form reminds me of Cardano's "cubic formula".
If you let x= a+ b, m= 3ab and then , a "reduced" cubic.
Suppose we know m and n. Can we solve for a and b and so find x? Yes, we can!
From m= 3ab, so . Multiplying through by gives the quadratic equation, in , or . By the quadratic formula, .
From , .
looks just like "a", above, with so and so that and m= 3.
That is, is a root of or . It is easy to see, say by graphing, that that equation has only one real root and, of course, so that single real root, and so the real value of the original expression, is 1.