Question

if x y z are natural numbers x2+xy+yz+zx = 30, y2+xy+yz+zx = 15, z2+xy+yz+zx = 18 , then x2 + y2 + z2 = _____________.

Answer 1

Let me help you with this query.
Given that,
x² + xy + yz + zx = 30 ........ (1)
y² + xy + yz + zx = 15 ........ (2)
z² + xy + yz + zx = 18  ....... (3)
We have to find,
x² + y² + z² = ?
Solution:
We know that,
(x + y + z) ² = x² + y² + z² + 2xy + 2yz + 2 xz  ........ (4)
Adding equations (1) , (2) and (3) , we get:
(x² + xy + yz + zx) + (y² + xy + yz + zx ) + (z² + xy + yz + zx) = 15 + 30 + 18
x² + y² + z² + 3 (xy + yz + xz) = 63
It can also be written as,
x² + y² + z² + 2 xy + 2yz + 2xz + (xy + yz + xz) = 63
x² + y² + z² + 2 xy + 2yz + 2xz = 63 - (xy + yz + xz)
 or (x + y + z) ² = 63 - (xy + yz + xz)    (From Eq. (4))
Hope this will help you. Thanks.