Question

If x and y are positive integers. If x4 + y4 + x2y2 = 481 and xy = 12,
then what is the value of x2 - xy + y2 ?​

Answer 1

Algebra

Given: x⁴ + y⁴ + x²y² = 481 and xy = 12

To find: x² - xy + y² = ?

Solution:

Here, x⁴ + y⁴ + x²y² = 481

or, (x² + y²)² - 2 x²y² + x²y² = 481

or, (x² + y²)² - x²y² = 481

or, (x² + y² + xy) (x² + y² - xy) = 481 .....(1)

Now, x² + y² + xy

= √{(x² + y²)²} + xy

= √(x⁴ + y⁴ + 2x²y²) + xy

= √(x⁴ + y⁴ + x²y² + x²y²) + xy

= √(481 + 12²) + 12, using given values

= √(481 + 144) + 12

= √(625) + 12

= 25 + 12

= 37

From (1), we get

37 (x² + y² - xy) = 481

or, x² + y² - xy = 481 / 37

or, x² + y² - xy = 13

Answer:

x² + y² - xy = 13