If the difference between the roots of the quadratic equation A x^2 − B x + C = 0 is 4, then which of the following relations between A, B and C is true?

Introduction / Context:
This question relates the coefficients of a quadratic equation to information about its roots. Knowing the difference between the roots gives a condition that must be satisfied by A, B and C. By using standard relationships between roots and coefficients, we derive the required relation and compare it to the given options.

Given Data / Assumptions:

• Quadratic equation: A x^2 − B x + C = 0, with A ≠ 0.
• Let the roots be r1 and r2.
• The difference between the roots is r1 − r2 = 4 or r2 − r1 = 4 (magnitude 4).
• We must find a correct relation involving A, B and C that encodes this information.

Concept / Approach:
For a quadratic equation A x^2 − B x + C = 0, the sum and product of roots are given by: r1 + r2 = B / A and r1 r2 = C / A. The squared difference of the roots can be written as (r1 − r2)^2 = (r1 + r2)^2 − 4 r1 r2. Since the magnitude of the difference is 4, we have (r1 − r2)^2 = 16. By substituting the expressions in terms of A, B and C, we obtain an equation relating these coefficients and then match it to one of the options.

Step-by-Step Solution:
1. Let r1 and r2 be the roots of A x^2 − B x + C = 0. 2. Then r1 + r2 = B / A. 3. Also, r1 r2 = C / A. 4. The given information says that the difference between roots has magnitude 4, so (r1 − r2)^2 = 16. 5. Use the identity (r1 − r2)^2 = (r1 + r2)^2 − 4 r1 r2. 6. Substitute in terms of coefficients: (r1 + r2)^2 = (B / A)^2 = B^2 / A^2. 7. Also 4 r1 r2 = 4 * (C / A) = 4C / A. 8. So (r1 − r2)^2 = B^2 / A^2 − 4C / A. 9. Set this equal to 16: B^2 / A^2 − 4C / A = 16. 10. Multiply both sides by A^2 to clear denominators: B^2 − 4AC = 16A^2. 11. Rearrange to B^2 − 16A^2 = 4AC. 12. Now look at the options and see which is equivalent to B^2 − 16A^2 = 4AC. 13. Option B is B^2 − 10A^2 = 4AC + 6A^2. 14. Simplify option B: B^2 − 10A^2 = 4AC + 6A^2 implies B^2 − 10A^2 − 4AC − 6A^2 = 0, or B^2 − 16A^2 − 4AC = 0. 15. This can be rewritten as B^2 − 16A^2 = 4AC, which matches our derived relation exactly.

Verification / Alternative check:
To confirm further, note that options A, C and D contain additional or different terms involving A^2 or B^2 that do not reduce to B^2 − 16A^2 = 4AC when simplified. Option B is the only one that directly rearranges to that clean equality. Therefore it must be the correct one.

Why Other Options Are Wrong:
Option A: B^2 − 16A^2 = 4AC + 4B^2 would simplify to B^2 − 16A^2 − 4AC − 4B^2 = 0 or −3B^2 − 16A^2 − 4AC = 0, which does not match B^2 − 16A^2 = 4AC.
Option C and option D similarly produce expressions where the coefficients of A^2 and B^2 do not reduce to the required relationship. None of them is algebraically equivalent to B^2 − 16A^2 = 4AC.

Common Pitfalls:
Students often confuse the formula for the difference between roots with that for the product or sum, or they incorrectly use 2√(discriminant) instead of working with squared difference directly. Another common error is failing to clear denominators properly, which leads to incorrect relations among A, B and C. Carefully applying the root coefficient identities and simplifying step by step avoids these issues.

Final Answer:
The correct relation is encoded by option B, which is equivalent to B^2 − 16A^2 = 4AC, so the answer is B^2 − 10A^2 = 4AC + 6A^2.