Asked by Anonymous
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
Answered by
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~ :)
There are no AI answers yet. The ability to request AI answers is coming soon!
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.