If a(x + y) = b(x − y) = 2ab, then what is the value of 2(x^2 + y^2) in terms of a and b? Derive x and y from the given equalities and simplify the final expression completely.

Introduction / Context:
This algebra question tests working with chained equalities and converting them into solvable equations. When you see a(x + y) = b(x − y) = 2ab, it means each expression equals the same common value (2ab). So you can create two separate equations: a(x + y) = 2ab and b(x − y) = 2ab. From these, you can find x + y and x − y directly, then solve for x and y. Finally, you compute x^2 + y^2 and multiply by 2. A powerful identity here is that x^2 + y^2 can be derived from (x + y)^2 and (x − y)^2, or by directly substituting x and y after solving. These problems are common in simplification because the arithmetic becomes clean once you interpret the equalities correctly.

Given Data / Assumptions:

• a(x + y) = 2ab • b(x − y) = 2ab • a and b are non-zero so division is valid • Required: 2(x^2 + y^2) in terms of a and b

Concept / Approach:
From a(x + y) = 2ab, divide by a to get x + y = 2b. From b(x − y) = 2ab, divide by b to get x − y = 2a. Then solve: x = ((x + y) + (x − y))/2 and y = ((x + y) − (x − y))/2. Compute x^2 + y^2 and multiply by 2. Simplification should lead to a clean expression in a^2 and b^2.

Step-by-Step Solution:
1) Since a(x + y) = 2ab, divide both sides by a: x + y = 2b 2) Since b(x − y) = 2ab, divide both sides by b: x − y = 2a 3) Add the two equations: (x + y) + (x − y) = 2b + 2a ⇒ 2x = 2(a + b) ⇒ x = a + b 4) Subtract the second from the first: (x + y) − (x − y) = 2b − 2a ⇒ 2y = 2(b − a) ⇒ y = b − a 5) Compute x^2 + y^2: x^2 + y^2 = (a + b)^2 + (b − a)^2 = (a^2 + 2ab + b^2) + (b^2 − 2ab + a^2) = 2a^2 + 2b^2 6) Multiply by 2: 2(x^2 + y^2) = 2(2a^2 + 2b^2) = 4(a^2 + b^2)

Verification / Alternative check:
Choose simple values a = 1, b = 2. Then x + y = 4 and x − y = 2. Solving gives x = 3 and y = 1. Now 2(x^2 + y^2) = 2(9 + 1) = 20. Formula gives 4(a^2 + b^2) = 4(1 + 4) = 20. Both match, confirming the result.

Why Other Options Are Wrong:
• 2(a^2 + b^2): misses the final multiplication by 2 after finding x^2 + y^2. • 4(a^2 − b^2) and 2(a^2 − b^2): come from sign mistakes while squaring (b − a)^2. • 8ab: incorrectly assumes x^2 + y^2 becomes a product instead of a sum of squares.

Common Pitfalls:
• Forgetting that the chained equality means two separate equations equal to 2ab. • Dividing by a or b without noting they must be non-zero. • Expanding (b − a)^2 incorrectly as b^2 − a^2 instead of b^2 − 2ab + a^2.

Final Answer:
4(a^2 + b^2)