Compare x and y (determine a single relation): I. 10x^2 − 7x + 1 = 0 II. 35y^2 − 12y + 1 = 0 Choose the correct relationship between x and y: x > y, x < y, x = y, or cannot be determined.

Difficulty: Medium

x > y
x < y
x = y
Relationship cannot be determined

Correct Answer: Relationship cannot be determined

Explanation:

Introduction / Context:
We compare any root x of the first quadratic with any root y of the second. If root intervals overlap, the relationship depends on the particular choice, making a universal statement impossible.

Given Data / Assumptions:

I: 10x^2 − 7x + 1 = 0.
II: 35y^2 − 12y + 1 = 0.

Concept / Approach:
Compute both roots from each quadratic using the quadratic formula. Then compare smallest/ largest values for overlaps.

Step-by-Step Solution:

I: Δ = (−7)^2 − 4*10*1 = 49 − 40 = 9 ⇒ √Δ = 3.x = [7 ± 3]/(20) ⇒ x ∈ {10/20 = 0.5, 4/20 = 0.2}.II: Δ = (−12)^2 − 4*35*1 = 144 − 140 = 4 ⇒ √Δ = 2.y = [12 ± 2]/(70) ⇒ y ∈ {14/70 = 0.2, 10/70 ≈ 0.142857}.x takes values {0.2, 0.5}; y takes {0.2, 0.142857…}. Depending on the picks, x can equal y (0.2), be greater (0.5 > any y), or be greater vs 0.142857 but equal vs 0.2. No universal strict inequality or equality holds.

Verification / Alternative check:
All 4 pairings: (0.2, 0.142857) ⇒ x > y; (0.5, 0.142857) ⇒ x > y; (0.2, 0.2) ⇒ x = y; (0.5, 0.2) ⇒ x > y.

Why Other Options Are Wrong:

x < y: Never occurs.
x = y: Occurs only for one pairing.
x > y: Fails in the equal case; not universal.

Common Pitfalls:
Ignoring that any root may be chosen, not just the larger or smaller one.

Final Answer:

Relationship cannot be determined

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Compare x and y (determine a single relation): I. 10x^2 − 7x + 1 = 0 II. 35y^2 − 12y + 1 = 0 Choose the correct relationship between x and y: x > y, x < y, x = y, or cannot be determined.

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