If xy = -30 and x^2 + y^2 = 61, then what is the value of (x + y)?

Introduction / Context:
This question uses algebraic identities that relate the sum and product of two numbers to the sum of their squares. Instead of solving for x and y separately, we use a known identity to compute x + y directly from the given information. This is a standard technique in aptitude and algebra problems.

Given Data / Assumptions:

• xy = -30.
• x^2 + y^2 = 61.
• We need the value of x + y.
• x and y are real numbers.

Concept / Approach:
We apply the identity (x + y)^2 = x^2 + 2xy + y^2. Since we know x^2 + y^2 and xy, we can substitute these values and find (x + y)^2. Then we take the square root to find x + y. Because the expression involves a square root, there can be two possible values, but we will choose the one that appears among the answer options.

Step-by-Step Solution:
Step 1: Start with the identity (x + y)^2 = x^2 + 2xy + y^2. Step 2: Substitute x^2 + y^2 = 61 and xy = -30. Step 3: Compute 2xy = 2 * (-30) = -60. Step 4: Then (x + y)^2 = 61 + (-60) = 1. Step 5: So x + y = ±√1 = ±1. Step 6: Among the given options, the value 1 appears, while -1 appears as a separate distractor. Step 7: The question asks for the value of x + y, and the valid value that matches an option is 1.

Verification / Alternative check:
We can attempt to find one pair (x, y) that satisfies the conditions and check the sum. Suppose x = 6 and y = -5. Then xy = 6 * (-5) = -30 and x^2 + y^2 = 36 + 25 = 61. So the conditions are satisfied. The sum x + y = 6 + (-5) = 1, which matches the selected value.

Why Other Options Are Wrong:

• Option A: 2 would give (x + y)^2 = 4, not 1, and would contradict x^2 + y^2 = 61.
• Option B: 3 would give (x + y)^2 = 9, not 1.
• Option D: 4 would imply (x + y)^2 = 16, which is inconsistent with the given data.
• Option E: -1 is mathematically possible as an abstract solution, but the example that satisfies all conditions gives x + y = 1 and the test typically expects the positive value when both are valid and the question does not emphasise sign choice.

Common Pitfalls:
Some learners forget to include the 2xy term in the identity or misuse x^2 + y^2 = (x + y)^2, which is incorrect. Others may not recognise that both positive and negative roots are possible when taking square roots. It is important to remember the correct identity and to examine which root makes sense in the context of given options or additional information.

Final Answer:
Hence, the value of (x + y) is 1.