System of quadratic equation with three variables

Question

Let $x, y, z$ be real numbers such that :

$$x^2+ xy+ yz+ zx = 30$$ $$y^2+ xy+ yz+ zx = 15$$ $$z^2+ xy+ yz+ zx = 18$$ ,

then: $$x^2+ y^2+ z^2 = ......$$.

Answer 1

It's $$(x+y)(x+z)=30,$$ $$(x+y)(y+z)=15$$ and $$(x+z)(z+y)=18,$$ which gives $$(x+y)^2(x+z)^2(y+z)^2=8100.$$

Now, we have two cases:

1. $$(x+y)(x+z)(y+z)=90,$$ which gives $$x+y=5,$$ $$x+z=6$$ and $$y+z=3,$$ which gives $x+y+z=7$ and $$(x,y,z)=(4,1,2)$$ and $$x^2+y^2+z^2=21.$$

The second case for you.