Recover correct roots from two partial mistakes: Two students solve x^2 + px + q = 0. Student A uses a wrong p and gets roots 2 and 6. Student B uses a wrong q and gets roots 2 and −9. Find the correct pair of roots.

Introduction / Context:
Each student changed only one coefficient relative to the true quadratic. From Student A’s roots (2, 6), we can fix the correct constant term since product depends on q only. From Student B’s roots (2, −9), we can fix the correct x-coefficient since sum depends on p only. With p and q recovered, the correct roots follow immediately.

Given Data / Assumptions:

• Correct equation: x^2 + px + q = 0
• A (wrong p): roots 2 and 6 ⇒ product = q = 12
• B (wrong q): roots 2 and −9 ⇒ sum = −p = 2 + (−9) = −7 ⇒ p = 7

Concept / Approach:
Use Vieta’s formulas separately on the two erroneous cases to extract the unchanged coefficient in each. Combine to obtain the true equation, then solve for its roots either by factoring or formula.

Step-by-Step Solution:

Verification / Alternative check:
Check: Using q = 12 with A’s wrong p produces the product 12; using p = 7 with B’s wrong q produces sum −7. Both align with the problem statement.

Why Other Options Are Wrong:

• 3 and −4; −3 and 4; 3 and 4: Do not satisfy x^2 + 7x + 12 = 0.

Common Pitfalls:
Assuming both p and q were wrong in the same attempt. The text specifies only one error per student.

Final Answer:
−3 and −4