Use (a^2 − b^2) = (a − b)(a + b): The difference of the squares of two numbers is 256000 and their sum is 1000. Find the two numbers.

Introduction / Context:This problem is a direct application of the identity a^2 − b^2 = (a − b)(a + b). With the sum given, the difference of the squares quickly yields the difference of the numbers, from which the pair is recovered uniquely (assuming a ≥ b).

Given Data / Assumptions:

• a^2 − b^2 = 256000
• a + b = 1000
• We look for real numbers a ≥ b; integers are suggested by the data.

Concept / Approach:Factor the square difference: (a − b)(a + b) = 256000. Since a + b = 1000, it follows that a − b = 256000 / 1000 = 256. Solve the simultaneous equations for a and b.

Step-by-Step Solution:a + b = 1000a − b = 256Add to get a: 2a = 1256 → a = 628Then b = 1000 − 628 = 372

Verification / Alternative check:Compute a^2 − b^2 = (a − b)(a + b) = 256 * 1000 = 256000, which matches exactly.

Why Other Options Are Wrong:600,400 gives (a − b) = 200, not 256; 640,360 gives (a − b) = 280; the others do not satisfy both the sum and square-difference conditions.

Common Pitfalls:Confusing a^2 − b^2 with (a − b)^2. The correct identity uses the product (a − b)(a + b).

Final Answer:628, 372