Divide 20 into two parts such that the sum of the squares of the two parts is 232. What are the two parts?

Introduction / Context:
This algebra question asks you to split the number 20 into two parts such that the sum of their squares equals 232. You must find the specific pair of numbers that satisfies both the sum condition and the sum of squares condition. This tests your ability to form and solve quadratic equations using basic identities.

Given Data / Assumptions:

- Let the two parts be x and y. - Their sum is x + y = 20. - The sum of their squares is x^2 + y^2 = 232. - We seek real number solutions, and the answer options suggest positive integers.

Concept / Approach:
We have two equations involving x and y: one for their sum and one for the sum of their squares. Using the identity (x + y)^2 = x^2 + y^2 + 2xy, we can express xy in terms of known quantities. Once we know x + y and xy, we can form a quadratic equation with roots x and y. Solving this quadratic yields the required two numbers. This method avoids guessing and ensures that both conditions are satisfied.

Step-by-Step Solution:
Step 1: Let x and y be the two parts with x + y = 20. Step 2: We are given x^2 + y^2 = 232. Step 3: Use the identity (x + y)^2 = x^2 + y^2 + 2xy. Step 4: Substitute the known values: (20)^2 = 232 + 2xy. Step 5: Compute 20^2 = 400. Step 6: The equation becomes 400 = 232 + 2xy. Step 7: Subtract 232 from both sides: 400 - 232 = 2xy, so 168 = 2xy. Step 8: Divide by 2 to find xy: xy = 84. Step 9: Now x and y are roots of the quadratic equation t^2 - (sum)t + (product) = 0, that is t^2 - 20t + 84 = 0. Step 10: Solve t^2 - 20t + 84 = 0 by factoring. We need two numbers whose product is 84 and whose sum is 20. Step 11: The pair 6 and 14 satisfies 6 * 14 = 84 and 6 + 14 = 20. Step 12: Therefore the two parts are 6 and 14.

Verification / Alternative check:
Verify both conditions with x = 6 and y = 14. The sum is 6 + 14 = 20, which matches the requirement. The sum of squares is 6^2 + 14^2 = 36 + 196 = 232, exactly as given in the problem. Therefore this pair satisfies both conditions perfectly and is the correct answer.

Why Other Options Are Wrong:
For 8 and 12, the sum is 20 but the sum of squares is 64 + 144 = 208, not 232. For 4 and 16, the sum is 20 but 4^2 + 16^2 = 16 + 256 = 272. For 10 and 10, the sum is 20 but 10^2 + 10^2 = 200. None of these match the required sum of squares of 232, so they are incorrect.

Common Pitfalls:
Some learners may try to guess pairs that add to 20 and check their squares one by one, which is inefficient and prone to error. Others may misuse the identity or forget the factor of 2 in 2xy, leading to wrong values for xy. Using the identity systematically and forming the quadratic equation ensures a clear and reliable solution path.

Final Answer:
The two parts into which 20 must be divided are 6 and 14, which corresponds to option A.