Question
Suppose x and y are real numbers such that xy = 9 and x2y + xy2 + x + y = 100. What is the integer value of x2 + y2? (Target Round #6)
Answers
Jai
observe that we can factor xy from (x^2)y + x(y^2):
(x^2)y + x(y^2) + x + y = 100:
xy [ x + y ] + x + y = 100
then we can factor (x + y):
(x + y)(xy + 1) = 100
since xy = 9,
(x + y)(9 + 1) = 100
(x + y)(10) = 100
x + y = 10
getting its square,
(x + y)^2 = 100
x^2 + 2xy + y^2 = 100
since xy = 9,
x^2 + y^2 + 2*9 = 100
x^2 + y^2 + 18 = 100
x^2 + y^2 = 82
hope this helps~ :)
(x^2)y + x(y^2) + x + y = 100:
xy [ x + y ] + x + y = 100
then we can factor (x + y):
(x + y)(xy + 1) = 100
since xy = 9,
(x + y)(9 + 1) = 100
(x + y)(10) = 100
x + y = 10
getting its square,
(x + y)^2 = 100
x^2 + 2xy + y^2 = 100
since xy = 9,
x^2 + y^2 + 2*9 = 100
x^2 + y^2 + 18 = 100
x^2 + y^2 = 82
hope this helps~ :)