Find the exact solutions of x^2 - (y - 5)^2 = 25 and y = -x^2
Do I try to isolate the y in the first equation first?
x = 0 is NOT a solution (if we have x = 0, that would lead to -25 = 25), and if it was, since y = -x^2, if x = 0, then y = 0, so a solution would be (0,0)
I'll start you off.
$\displaystyle x^2 - \left( -x^2 - 5 \right)^2 = 25$
$\displaystyle \Rightarrow x^2 - \left( x^4 + 10x^2 + 25 \right) = 25$
$\displaystyle \Rightarrow x^2 - x^4 - 10x^2 - 25 = 25$
Now continue (it seems as though the solutions are complex, as in, not real)
Note that we can rewrite this as: $\displaystyle x^4 + 9x^2 + 50 = 0$
which in turn we could rewrite as: $\displaystyle \left( x^2 \right)^2 + 9 \left( x^2 \right) + 50 = 0$
which is a quadratic eqaution in $\displaystyle x^2$, we can therefore employ the quadratic formula