Common roots across two quadratics: I. x^2 − 19x + 84 = 0 II. y^2 − 25y + 156 = 0 How many integer values are common to the two root-sets?

Introduction / Context:
Instead of comparing unknowns, we repair the ambiguous stem to a precise and valuable task: count the common integer roots of two given quadratics. Factoring both equations quickly reveals the root sets and their intersection size.

Given Data / Assumptions:

• I: x^2 − 19x + 84 = 0
• II: y^2 − 25y + 156 = 0
• We count how many integers appear in both sets of roots.

Concept / Approach:
Factor each quadratic: for t^2 − St + P = 0, find two integers whose sum is S and product is P. The solutions are those integers. Then take the intersection of the two sets and count elements.

Step-by-Step Solution:

Verification / Alternative check:
Check by substitution: each candidate in the intersection should satisfy both quadratics. 12 satisfies both; 7 and 13 are unique to one each.

Why Other Options Are Wrong:
2, 3, 4, 0 miscount the intersection size; only one integer (12) is shared.

Common Pitfalls:
Arithmetic slips while factoring, or missing that “how many” asks for count of shared values, not listing all roots.

Final Answer:
1