Solving a system of real-valued polynomials

Question

How would i solve this system of equations

$(x+y+z)^2 - 4 - 5yz = 0$

$(x+y+z)^2 - 4 - 5xz = 0$

$(x+y+z)^2 - 4 - 5xy = 0$

Answer 1

hint: Subtract any two of the equations: $z(x-y) = 0 = y(x-z) = x(y-z)$. Can you take it from here?

Answer 2

In general, solving systems of polynomials is a very hard problem—so hard that two whole fields, algebraic geometry and commutative algebra, have grown up around it. If you want to solve an arbitrary system of polynomials, you're probably going to need heavy-duty tools like Gröbner bases.

Your system of polynomials, however, is far from arbitrary. For one thing, the expression $(x + y + z)^2 - 4$ appears in every equation! When an expression shows up several times, it's often useful to make up a name for it, so try creating a new variable $t = (x + y + z)^2 - 4$ and substituting it into all the equations. This may help you see what to do next.

(This hint is actually the same as A1DHTH's hint, but from a different point of view.)