SOLUTION: Find two numbers such that their sum multiplied by the sum of their squares is 65, and the difference multiplied by the difference of their square is 5.

Question 909164: Find two numbers such that their sum multiplied by the sum of their squares is 65, and the difference multiplied by the difference of their square is 5.
Answer by mananth(16946) (Show Source): You can put this solution on YOUR website!
(x+y)(x^2+y^2)=65
x(x^2+y^2)+y(x^2+y^2)=65
x^3+xy^2+x^2y+y^3=65..................(1)
(x-y)(x^2-y^2)=5
x(x^2-y^2)-y(x^2-y^2)=5
x^3-xy^2-x^2y+y^3=5.....................(2)
add (1) & (2)
2x^3+2y^3=70
x^3+y^3=35.........................(3)
subtract (2) from (1)
2x^2y+2xy^2=60
2xy(x+y)=60
xy(x+y)=30
from (3)
x^3+y^3=35.........................(3)
(x+y)^3-3xy(x+y)=35
(x+y)^3+3(30)=35
(x+y)^3=90+35
(x+y)^3=125
(x+y)=5...........................(4)
Now we know
(x-y)(x^2-y^2)=5
(x-y)(x+y)(x-y)=5
(x+y)(x-y)^2=5
5(x-y)^2=5
(x-y)^2=5/5=1
x-y=+/-1...................................(5)
solve (4)&(5)

we have (2,3) OR (3,2) as the solutions

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