f x and y are integers and 2x + 9y = 57, then the least possible value of x^2 + 2xy + y^2 is:?

Suraj Pandey

Understand the problem
We need to minimize x^2 + 2xy + y^2 given 2x + 9y = 57.

Express y in terms of x
2x + 9y = 57
y = (57 - 2x) / 9

Substitute y in the expression to be minimized
f(x, y) = x^2 + 2xy + y^2

Find integer solutions for y
x must be chosen such that (57 - 2x) / 9 is an integer.
Test values for x:
  x = 3: y = (57 - 6) / 9 = 51 / 9 = 5.67 (not an integer)
  x = 12: y = (57 - 24) / 9 = 33 / 9 = 3.67 (not an integer)
  x = 21: y = (57 - 42) / 9 = 15 / 9 = 1.67 (not an integer)
  x = 30: y = (57 - 60) / 9 = -3 / 9 = -1/3 (not an integer)
  x = 39: y = (57 - 78) / 9 = -21 / 9 = -2.33 (not an integer)

Verify integer values for y
Find values of x such that y is an integer:
 x = -3, y = 7

Compute x^2 + 2xy + y^2
For x = -3, y = 7:
f(-3, 7) = (-3)^2 + 2(-3)(7) + (7)^2
= 9 - 42 + 49
= 16

Conclusion
The least possible value of x^2 + 2xy + y^2 is 16.