Divide 32 into two real parts such that the sum of the squares of the two parts is 674. What are the values of the two parts?

Introduction / Context:
This algebra problem asks you to split a given number into two parts under a condition involving the sum of their squares. You must divide 32 into two parts such that the sum of the squares of these parts is 674. The question tests quadratic equation skills and understanding of how to model number splitting.

Given Data / Assumptions:
- The total of the two parts is 32.
- Let the two parts be x and 32 minus x.
- The sum of the squares of these parts equals 674.
- The parts are real numbers, and the options indicate integer solutions.

Concept / Approach:
When a number is split into two parts, it is natural to represent one part as x and the other as 32 minus x. The condition on the sum of squares becomes an equation in x: x^2 + (32 - x)^2 = 674. Expanding and simplifying this expression results in a quadratic equation that can be solved using factorization.

Step-by-Step Solution:
Step 1: Let the first part be x. Then the second part is 32 - x because their sum is 32.Step 2: The sum of the squares is given by x^2 + (32 - x)^2.Step 3: According to the question, x^2 + (32 - x)^2 = 674.Step 4: Expand (32 - x)^2 to get 1024 - 64x + x^2.Step 5: Substitute back: x^2 + 1024 - 64x + x^2 = 674 which simplifies to 2x^2 - 64x + 1024 = 674.Step 6: Subtract 674 from both sides: 2x^2 - 64x + 350 = 0.Step 7: Divide by 2 to simplify: x^2 - 32x + 175 = 0.Step 8: Factorize: x^2 - 32x + 175 = (x - 25)(x - 7) = 0, so x = 25 or x = 7.Step 9: Thus the two parts are 25 and 7, since 25 + 7 = 32.

Verification / Alternative check:
Check the condition for both possible assignments. If x = 25, the other part is 7. The sum of squares is 25^2 + 7^2 = 625 + 49 = 674, which matches the requirement. If x = 7 and the other part is 25, the sum is the same. So the unordered pair of parts is {25, 7}, consistent with the options given.

Why Other Options Are Wrong:
Option 22 and 10 gives squares 484 and 100, sum 584. Option 30 and 2 gives 900 and 4, sum 904. Option 20 and 12 gives 400 and 144, sum 544. Option 18 and 14 gives 324 and 196, sum 520. None of these sums equal 674, so they do not satisfy the condition.

Common Pitfalls:
Sometimes learners misrepresent the second part as 32 + x instead of 32 - x, which leads to an incorrect equation. Others make expansion errors when squaring 32 - x. Forgetting to divide by 2 when simplifying the quadratic or making mistakes while factorizing can also cause wrong answers. Being systematic in algebraic steps helps avoid these issues.

Final Answer:
The required parts into which 32 should be divided are 25 and 7.