Answer: Given the two equations x + y = 8 and x^2 + y^2 = 80, we can use the first equation to solve for x or y and substitute that expression into the second equation.
Let's solve for y:
y = 8 - x
Now we can substitute this expression for y into the second equation:
x^2 + (8 - x)^2 = 80
Expanding the square and collecting terms gives:
x^2 + 64 - 16x + x^2 = 80
Combining like terms:
2x^2 - 16x + 16 = 80
Moving all terms to the left side:
2x^2 - 16x - 64 = 0
Using the quadratic formula, we can solve for x:
x = (-b ± √(b^2 - 4ac)) / (2a)
where a = 2, b = -16, and c = -64
x = (-(-16) ± √((-16)^2 - 4 * 2 * -64)) / (2 * 2)
x = (16 ± √(256 + 512)) / 4
x = (16 ± √768) / 4
x = (16 ± 8√6) / 4
Since x and y must be real numbers, we reject the negative square root. Therefore, x = (16 + 8√6) / 4.
Finally, substituting x into the first equation to find y:
y = 8 - x = 8 - (16 + 8√6) / 4 = (8 - 16) / 4 - 8√6 / 4 = -8 / 4 - 8√6 / 4 = -2 - 2√6.
Therefore, xy = x * y = (16 + 8√6) / 4 * (-2 - 2√6) = -16 - 16√6.
The answer is d. -16.
Explanation: