Let x be the greater real root of x^2 − 365 = 364 and y satisfy y − √324 = √81. Compare x and y.

Difficulty: Easy

If x > y
If x ≥ y
If x < y
If x ≤ y
If x = y

Correct Answer: If x ≥ y

Explanation:

Introduction / Context:
We compare exact numerical values derived from simple square roots. The equations are arranged to yield integers after evaluation, simplifying the comparison.

Given Data / Assumptions:

x^2 − 365 = 364 ⇒ x^2 = 729 ⇒ roots ±27; greater x = 27.
y − √324 = √81 ⇒ y − 18 = 9 ⇒ y = 27.

Concept / Approach:
Evaluate each side carefully. √324 = 18 and √81 = 9. Then solve the simple linear relation for y and compare with x.

Step-by-Step Solution:

x = 27.y = 27.Therefore x = y, which implies x ≥ y (and also x ≤ y).

Verification / Alternative check:
Substitutions confirm both equations are satisfied by 27.

Why Other Options Are Wrong:
Strict inequalities do not hold since equality is exact.

Common Pitfalls:
Misreading the radicals or arithmetic around subtracting and adding 18 and 9.

Let x be the greater real root of x^2 − 365 = 364 and y satisfy y − √324 = √81. Compare x and y.

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