Find $x, y$ and $z$ such that $x+y+z=23$, $xy+yz+xz=144$ and $xyz=252$, where $x\gt y\gt z$.

Question

All I can think of is that $x$, $y$ and $z$ are the roots of the multiples of polynomial $P(x)=x^3-23x^2+144x-252$, and I found the roots by rational root test.

However this method is not elegant IMO because if the roots are not rational, I have no idea how to find them anymore.

So, could anyone introduce me a general method or at least a better method in solving system equations such as $x+y+z=a$, $xy+yz+xz=b$ and $xyz=c$, where $a, b, c\in \mathbb R$?

Answer 1

You can apply this identity:

$$(ab+bc+ca)(a+b+c)-abc=(a+b)(b+c)(c+a)$$

Putting values you get:

$(x+y)(y+z)(z+x) =3060$

Now you have to decompose 3060 and equate factors , you will have some system of equations to solve.