Question

X³+3xy²=14
Y³+3yx²=13
X²+y²=?

Answer 1

x³ + 3 x y² = 14      --- (1)
y³ + 3 x² y = 13    ---- (2)

Add  (1) and (2)  =>  (x + y)³ = 14+ 13 = 3³
     =>  x + y = 3        --- (3)

Subtract (2) from (1) :  x³ - y³ + 3 x y² - 3 x² y = 14 - 13 = 1
     =>  (x - y) (x² + x y + y²) + 3 x y (y - x) = 1
     =>  (x - y) (x² + x y + y² - 3x y) = 1
     => (x - y) (x - y)² = 1
     => x - y = 1        ---(4)

from (3) and (4)  we get:  x = 2  and y = 1.

so x² + y² = 5.