If x + y = 10 and xy = 4 for real numbers x and y, then what is the value of x^4 + y^4?

Introduction / Context:
This is a standard algebraic identity question where the sum and product of two numbers x and y are given, and you are asked to find a higher power expression, namely x^4 + y^4. Instead of solving for x and y individually, which may involve solving a quadratic equation, we can use identities for powers in terms of x + y and xy. This approach is faster and less error prone.

Given Data / Assumptions:

• x + y = 10.
• xy = 4.
• x and y are real numbers.
• We need to find x^4 + y^4.

Concept / Approach:
We use the identity for squares and fourth powers. First, we find x^2 + y^2 from the relation (x + y)^2 = x^2 + y^2 + 2xy. Once we have x^2 + y^2, we use another identity for x^4 + y^4 in terms of (x^2 + y^2)^2 and x^2 y^2. Specifically, x^4 + y^4 = (x^2 + y^2)^2 − 2x^2 y^2. Since xy is known, x^2 y^2 is simply (xy)^2.

Step-by-Step Solution:
1. Start with (x + y)^2 = x^2 + y^2 + 2xy. 2. Substitute x + y = 10 and xy = 4. 3. Compute (x + y)^2: (10)^2 = 100. 4. So 100 = x^2 + y^2 + 2 * 4. 5. This becomes 100 = x^2 + y^2 + 8. 6. Therefore x^2 + y^2 = 100 − 8 = 92. 7. Now apply the identity x^4 + y^4 = (x^2 + y^2)^2 − 2x^2 y^2. 8. Compute x^2 y^2: since xy = 4, x^2 y^2 = (xy)^2 = 4^2 = 16. 9. Compute (x^2 + y^2)^2: 92^2 = 8464. 10. Substitute into the identity: x^4 + y^4 = 8464 − 2 * 16. 11. Simplify: 2 * 16 = 32. 12. So x^4 + y^4 = 8464 − 32 = 8432.

Verification / Alternative check:
We can verify the result by solving for x and y explicitly. x and y are roots of t^2 − (x + y)t + xy = 0, so t^2 − 10t + 4 = 0. Solving gives two real roots. If we compute the fourth powers of these roots and add them, we will obtain the same value, 8432. Although more tedious, this confirms that our identity based method is correct.

Why Other Options Are Wrong:
Option 1 (8464) corresponds to (x^2 + y^2)^2 without subtracting 2x^2 y^2, so it ignores the second part of the identity for x^4 + y^4.
Option 3 (7478) and option 4 (6218) have no natural connection to the algebraic identities derived from the given sum and product. They result from incorrect manipulation or random errors in calculation, such as subtracting 2xy instead of 2x^2 y^2.

Common Pitfalls:
Learners sometimes mistakenly use x^4 + y^4 = (x^2 + y^2)^2 instead of (x^2 + y^2)^2 − 2x^2 y^2. Others confuse the identity for x^3 + y^3 with that for x^4 + y^4, or they miscalculate 92^2. Careful recall of the correct algebraic formulas and methodical arithmetic steps prevent such mistakes.

Final Answer:
The value of x^4 + y^4 is 8432.