Find the common root of the quadratics x^2 − 7x + 10 = 0 and x^2 − 10x + 16 = 0.

Introduction / Context:
Two quadratics are given. A common root is a value of x that satisfies both equations. Factoring each quadratic allows direct identification of roots and their intersection.

Given Data / Assumptions:

• x^2 − 7x + 10 = 0
• x^2 − 10x + 16 = 0

Concept / Approach:
Factor both polynomials into linear factors and list their roots. The common root is the value appearing in both root sets.

Step-by-Step Solution:
x^2 − 7x + 10 = (x − 5)(x − 2) ⇒ roots: {5, 2}x^2 − 10x + 16 = (x − 8)(x − 2) ⇒ roots: {8, 2}Common root = 2

Verification / Alternative check:
Plug x = 2 into both equations: 4 − 14 + 10 = 0 and 4 − 20 + 16 = 0. Both equal zero, confirming.

Why Other Options Are Wrong:
−2, 3, 5, 8 are not roots common to both equations; only 2 appears in both sets.

Common Pitfalls:
Arithmetic slips in factoring; check factor pairs that sum to the middle coefficient and multiply to the constant term.

Final Answer:
2