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Compare x and y (determine a single relation): I. x^2 − 6x = 7 II. 2y^2 + 13y + 15 = 0 Choose the correct relationship between x and y: x > y, x < y, x = y, or cannot be determined.

Difficulty: Medium

Correct Answer: x > y

Explanation:


Introduction / Context:
We must compare any root from each equation. Solving both gives finite root sets; comparing their ranges yields the universal relation if one exists.

Given Data / Assumptions:

  • I: x^2 − 6x − 7 = 0 (rearranged).
  • II: 2y^2 + 13y + 15 = 0.


Concept / Approach:
Solve both quadratics. If the smallest x exceeds the largest y, then x > y for all pairings.

Step-by-Step Solution:

I: x^2 − 6x − 7 = 0 ⇒ Δ = 36 + 28 = 64 ⇒ x = [6 ± 8]/2 ⇒ x ∈ {7, −1}.II: Δ = 13^2 − 4*2*15 = 169 − 120 = 49 ⇒ √Δ = 7.y = [−13 ± 7]/(4) ⇒ y ∈ { (−6/4)=−1.5 , (−20/4)=−5 }.Hence x ∈ {−1, 7} and y ∈ {−1.5, −5}. The smallest x is −1 and the largest y is −1.5. Since −1 > −1.5 and 7 > any y, every x is greater than every y.


Verification / Alternative check:
Check all four pairings; each gives x − y > 0.


Why Other Options Are Wrong:

  • x < y or x = y: Contradicted by direct computation.
  • Relationship cannot be determined: Incorrect; ordering is strictly separated.


Common Pitfalls:
Missing the sign when taking square roots or forgetting to rearrange to standard form before solving.


Final Answer:

x > y
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