Question

Calculati
a) x2+y2, stiind ca x+y=10 si xy=22
b) x2+y2, stiind ca x-y=2√2 si xy=47
c) x2+1 supra x2, stiind ca x+1/x=2√3
d) x2+1/x2, stiind ca x-1/x=6
e) x2+y2+z2, stiind ca x+y+z=7 si xy+xz+yz=10
f) x2+y2+z2, stiind ca x+y-z=7 si xy-xz-yz=10
g) x3+y3, stiind ca x+y=8 si xy=15

Answer 1

a) x²+y², stiind ca x+y=10 si xy=22
(x+y)²=x²+2xy+y²=10²=100
x²+y²=100-2xy=100-2*22=100-44=56

b) x²+y², stiind ca x-y=2√2 si xy=47
(x+y)²=x²-2xy+y²=(2√2)²=8
x²+y²=8+2xy=8+2*47=8+94=102

e) x²+y²+z², stiind ca x+y+z=7 si xy+xz+yz=10
(x+y+z)²=x²+y²+z²+2xy+2xz+2yz=x²+y²+z²+2(xy+xz+yz)=7+2*10=27

f) x²+y²+z², stiind ca x+y-z=7 si xy-xz-yz=10
(x+y-z)²=x²+y²+z²+2xy-2xz-2yz=x²+y²+z²+2(xy-xz-yz)=7+2*10=27

g) x³+y³, stiind ca x+y=8 si xy=15
(x+y)³=x³+3x²y+3xy²+y³ = x³+y³+3xy(x+y) =8³=512
x³+y³=512-3*15*8=512-360=152