Let α and β be the roots of a quadratic equation such that α + β = 8 and α − β = 2√5. Which of the following quadratic equations has roots α^4 and β^4?

Introduction / Context:
This problem is about transforming roots of a quadratic equation. Instead of directly solving for α and β, we are given their sum and difference and asked to build a new quadratic equation whose roots are α^4 and β^4. The task therefore involves manipulating symmetric expressions in α and β and applying identities for powers and sums.

Given Data / Assumptions:

• α and β are roots of some quadratic.
• α + β = 8.
• α − β = 2√5.
• We must find the quadratic equation with roots α^4 and β^4.
• The new quadratic will have the form x^2 − Sx + P = 0, where S = α^4 + β^4 and P = α^4 β^4.

Concept / Approach:
From the given sum and difference of α and β, we can compute α^2 and β^2 using the identities: α^2 = [(α + β)^2 + (α − β)^2] / 4 and β^2 = [(α + β)^2 − (α − β)^2] / 4. Once α^2 and β^2 are known, α^4 and β^4 follow by squaring. The sum S = α^4 + β^4 and product P = α^4 β^4 provide the coefficients of the required quadratic: x^2 − Sx + P = 0. We then match this with the options.

Step-by-Step Solution:
1. Compute (α + β)^2: (α + β)^2 = 8^2 = 64. 2. Compute (α − β)^2: (α − β)^2 = (2√5)^2 = 4 * 5 = 20. 3. Use α^2 = [(α + β)^2 + (α − β)^2] / 4. 4. So α^2 = (64 + 20) / 4 = 84 / 4 = 21. 5. For β^2, use β^2 = [(α + β)^2 − (α − β)^2] / 4. 6. Thus β^2 = (64 − 20) / 4 = 44 / 4 = 11. 7. Now compute α^4 and β^4: α^4 = (α^2)^2 = 21^2 = 441, and β^4 = (β^2)^2 = 11^2 = 121. 8. Sum of the new roots S = α^4 + β^4 = 441 + 121 = 562. 9. Product of the new roots P = α^4 β^4 = (21^2)(11^2) = (21 * 11)^2 = 231^2 = 53361. 10. Therefore the quadratic with roots α^4 and β^4 is x^2 − Sx + P = 0. 11. Substitute S and P: x^2 − 562x + 53361 = 0. 12. Compare this with the given options; it matches option A exactly.

Verification / Alternative check:
As an additional check, we can verify that the discriminant of this new equation is positive, guaranteeing real roots for x when α and β are real. The discriminant is Δ = S^2 − 4P = 562^2 − 4 * 53361. Computing this shows Δ is positive, which is consistent with x taking real values corresponding to α^4 and β^4 for real α and β. Although not strictly required for choosing the correct option, this supports the internal consistency of the result.

Why Other Options Are Wrong:
Option B: x^2 − 562x + 14641 = 0 uses the correct sum S = 562 but an incorrect product P, since we have already established P = 53361, not 14641.
Option C: x^2 − 64x + 14641 = 0 uses a sum of 64, confusing S with (α + β)^2 instead of α^4 + β^4.
Option D: x^2 − 64x + 53361 = 0 again misuses 64 as the coefficient of x while having the correct product P, so it does not correspond to the required pair of roots α^4 and β^4.

Common Pitfalls:
A common error is to assume the new sum of roots is (α + β)^4 or to confuse α^4 + β^4 with (α^2 + β^2)^2. Another frequent mistake is to forget that the product α^4 β^4 equals (αβ)^4, but here we correctly used α^2 and β^2 individually. Handling the arithmetic with squares, especially 231^2, also needs care. Working systematically with intermediate squares and double checking calculations helps avoid these traps.

Final Answer:
The quadratic equation whose roots are α^4 and β^4 is x^2 − 562x + 53361 = 0.