If x+y+z=16 and xy+yz+zx=85, what is x^2+y^2+z^2?
To find the value of x^2 + y^2 + z^2, we need to use the given equations and apply some algebraic manipulations.
We have the following two equations:
1) x + y + z = 16
2) xy + yz + zx = 85
To find x^2 + y^2 + z^2, let's first square equation 1:
(x + y + z)^2 = 16^2
(x^2 + y^2 + z^2) + 2(xy + yz + zx) = 256
x^2 + y^2 + z^2 + 2(xy + yz + zx) = 256
Now, let's substitute equation 2 into the above equation:
x^2 + y^2 + z^2 + 2(85) = 256
x^2 + y^2 + z^2 + 170 = 256
x^2 + y^2 + z^2 = 256 - 170
x^2 + y^2 + z^2 = 86
Therefore, x^2 + y^2 + z^2 is equal to 86.