China | Math Olympiad | A Very Nice Algebra Problem

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[@SidneiMV]

x² + y² = 13
x³ + y³ = 19
find E = x + y
(x + y)² = 13 + 2xy
xy = [(x + y)² - 13]/2
(x + y)³ = 19 + 3xy(x + y)
(x + y)³ = 19 + (3/2)[(x + y)² - 13](x + y)
(x + y)³ = 19 + (3/2)(x + y)³ - 39/2(x + y)
(1/2)(x + y)³ - 39/2(x + y) + 19 = 0
E³ - 39E + 38 = 0
E³ - E - 38(E - 1) = 0
E(E + 1)(E - 1) - 38(E - 1) = 0
(E - 1)(E² + E - 38) = 0
*E = 1*
E² + E - 38 = 0
*E = (-1 ± √153)/2*

[@comdo777]

asnwer=11 isit

[@comdo777]

asnwer=1+152/2 isit