8th grade math

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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)

  • 8th grade math -

    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~ :)

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