If xy = −18 and x² + y² = 85 for real numbers x and y, find the positive value of the sum x + y, that is, |x + y|.

Introduction / Context:
This question involves symmetric expressions in two variables, x and y. It tests your ability to connect x² + y² and xy with the square of the sum x + y. Such relationships are very common in algebra and aptitude exams, and mastering them allows you to avoid solving separate equations for x and y while still finding useful combinations like x + y or x − y.

Given Data / Assumptions:
- x and y are real numbers.
- xy = −18.
- x² + y² = 85.
- We are asked for the positive value of x + y, that is, |x + y|.

Concept / Approach:
The standard identity relating the square of a sum to the sum of squares and the product is:
(x + y)² = x² + y² + 2xy.
We are given x² + y² and xy, so we can directly compute (x + y)². Once we have that, we take the square root to find the magnitude of x + y. Because the problem asks for the positive value or absolute value, we choose the positive root.

Step-by-Step Solution:
Step 1: Write the identity (x + y)² = x² + y² + 2xy. Step 2: Substitute the given values x² + y² = 85 and xy = −18. Step 3: Compute 2xy = 2 * (−18) = −36. Step 4: Evaluate (x + y)² = 85 + (−36) = 85 − 36 = 49. Step 5: Therefore (x + y)² = 49. Step 6: Take square roots to find x + y. This gives x + y = 7 or x + y = −7. Step 7: The question explicitly asks for the positive value of the sum, which is |x + y|, so the required answer is 7.
Verification / Alternative check:
We can construct a pair of numbers that satisfy the conditions to check. Suppose x + y = 7 and xy = −18. These would be roots of the quadratic t² − 7t − 18 = 0. Solve this equation: the discriminant is 7² − 4 * 1 * (−18) = 49 + 72 = 121. The roots are (7 ± 11) / 2, so t = 9 or t = −2. Thus one possible pair is x = 9, y = −2. For this pair, x² + y² = 81 + 4 = 85 and xy = −18, exactly matching the problem. This confirms the use of the identity and the correctness of |x + y| = 7.

Why Other Options Are Wrong:
Option a, 8, would give (x + y)² = 64 and then x² + y² = (x + y)² − 2xy = 64 − 2 * (−18) = 64 + 36 = 100, not 85.
Option b, 10, yields x² + y² = 100 − 2 * (−18) = 136, which is incorrect.
Option c, 9, yields x² + y² = 81 − 2 * (−18) = 117, also incorrect.
Option e, 5, gives x² + y² = 25 + 36 = 61, again not equal to 85.

Common Pitfalls:
A common mistake is to forget that (x + y)² involves 2xy, not xy, leading to an incorrect computation. Some students also neglect the possibility of both positive and negative roots, but in this question the stem specifies that the positive value or absolute value is required, so the negative root is discarded. Another error is to attempt to find x and y individually, which is not necessary and can be more time consuming. Using the identity directly is efficient and reliable.

Final Answer:
The positive value of x + y, that is |x + y|, is 7.