If x = √5 + 1 and y = √5 - 1, then what is the value of (x^2 / y^2) + (y^2 / x^2) + 4[(x / y) + (y / x)] + 6?

Introduction / Context:
This problem involves symmetric expressions built from two conjugate surds, √5 + 1 and √5 - 1. Such expressions are common in algebra and aptitude problems, because conjugate pairs often lead to nice simplifications when multiplied or added. Here, you must evaluate a combination of ratios and squares involving x and y, which looks complicated at first glance but simplifies significantly once key relationships between x and y are identified.

Given Data / Assumptions:

• x = √5 + 1.
• y = √5 - 1.
• We need to evaluate E = (x^2 / y^2) + (y^2 / x^2) + 4[(x / y) + (y / x)] + 6.
• All roots are real, and denominators x and y are nonzero.

Concept / Approach:
Because x and y are conjugates, their product and sum take especially simple forms. Specifically, x y = (√5 + 1)(√5 - 1) and x^2, y^2 will also be related in a symmetric way. We can first compute x y, x / y and y / x and then express x^2 / y^2 and y^2 / x^2 in terms of these ratios. Another strategy is to set t = x / y + y / x and then note that x^2 / y^2 + y^2 / x^2 equals t^2 - 2, simplifying the overall expression in terms of t.

Step-by-Step Solution:
Step 1: Compute x y = (√5 + 1)(√5 - 1) = (5 - 1) = 4. Step 2: Compute x^2 and y^2. For x^2: (√5 + 1)^2 = 5 + 2√5 + 1 = 6 + 2√5. For y^2: (√5 - 1)^2 = 5 - 2√5 + 1 = 6 - 2√5. Step 3: Compute x / y. Since x / y = (√5 + 1) / (√5 - 1), multiply numerator and denominator by the conjugate (√5 + 1) to get (√5 + 1)^2 / (5 - 1) = (6 + 2√5) / 4 = (3 + √5) / 2. Step 4: Similarly, y / x equals (√5 - 1) / (√5 + 1). Rationalising gives (√5 - 1)^2 / 4 = (6 - 2√5) / 4 = (3 - √5) / 2. Step 5: Add the ratios: x / y + y / x = [(3 + √5) + (3 - √5)] / 2 = 6 / 2 = 3. Step 6: Compute t = x / y + y / x = 3. Then x^2 / y^2 + y^2 / x^2 equals t^2 - 2 = 3^2 - 2 = 9 - 2 = 7. Step 7: Substitute into E: E = 7 + 4t + 6 = 7 + 4 × 3 + 6 = 7 + 12 + 6 = 25.

Verification / Alternative check:
You can verify the result numerically by approximating √5 as about 2.236. Then x ≈ 3.236 and y ≈ 1.236. Compute x / y, y / x, x^2 / y^2 and y^2 / x^2 numerically and combine them using the given formula. The total will be extremely close to 25, confirming that the algebraic simplification is correct. This provides a useful sanity check when dealing with surds.

Why Other Options Are Wrong:

• Option a (31) and option e (19) are the results of arithmetic mistakes, such as squaring t incorrectly or miscomputing t^2 - 2.
• Options b and c involving multiples of √5 do not match the structure of the expression, which simplifies to a rational integer.
• Only 25 is consistent with both the algebraic derivation and numerical approximation.

Common Pitfalls:
A common error is to attempt to compute each term separately using approximate decimals without noticing the symmetry between x and y, which can introduce rounding errors and unnecessary complexity. Another pitfall is to misapply identities when relating x / y and y / x to x^2 / y^2 and y^2 / x^2. Using the substitution t = x / y + y / x and the identity t^2 = x^2 / y^2 + 2 + y^2 / x^2 offers a cleaner and more reliable route to the final answer.

Final Answer:
The value of the given expression is 25.