So the question asks to solve for real valued $a$ such that $b,c,d\in\mathbb{R}$ $$abcd=-1$$ $$(a+c)(b+d)=-1$$ $$ac+bd+a+b+c+d=-1$$ $$ab+cd=ac+a+c$$ So assuming the four numbers are roots of a quartic equation $x^4+b_3x^3+b_2x^2+b_1x+b_0=0$. How do we use Vieta relation or some other method to solve this ?
I could not get all the $b_i$ using the above equations without solving $a,b,c,d$ indivdually. Any ideas? Thanks
Curiously, this system turns out to involve $x^{17}-1=0$.
You have four equations in four unknowns. Resolving this to a single equation, we get two non-trivial octics, one with all real roots, the other with all complex roots. The relevant roots are,
$$\begin{aligned} a_k &=2\cos\Bigl(\frac{2k\,\pi}{17}\Bigr)\\ b_k &=2\cos\Bigl(\frac{4k\,\pi}{17}\Bigr)\\ c_k &=2\cos\Bigl(\frac{8k\,\pi}{17}\Bigr)\\ d_k &=2\cos\Bigl(\frac{16k\,\pi}{17}\Bigr) \end{aligned}$$
for $k = 1,2,3,\dots 8$. As a quartic, all these have the same form, namely,
$$-1 + (2 \mp \sqrt{17}) a + \tfrac{1}{2} (-3 \pm \sqrt{17}) a^2 + \tfrac{1}{2} (1 \pm \sqrt{17}) a^3 + a^4=0\tag1$$
and similarly for $b,c,d$.
P.S. It seems doubtful one can exclusively find $(1)$ without solving for $a,b,c,d$ individually, considering there is another quartic which yields valid but non-real solutions.