Multiple choice

Directions: In the following question, two equations numbered I and II are given. You have to solve both the equations and find out the relationship between x and y. I. 3x2 - 14x + 15 = 0 II. 6y2 - 5y + 1 = 0

  1. x > y

  2. x ≥ y

  3. x < y

  4. x ≤ y

  5. x = y or no relation can be established
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A Correct answer
Explanation

I: 3x^2 - 14x + 15 = 0 -> (3x-5)(x-3) = 0 -> x = 5/3 (1.66), 3. II: 6y^2 - 5y + 1 = 0 -> (3y-1)(2y-1) = 0 -> y = 1/3 (0.33), 1/2 (0.5). Since all x values are greater than all y values, x > y.

AI explanation

Factoring equation I, 3x^2 - 14x + 15 = 0, by splitting the middle term gives 3x^2 - 9x - 5x + 15 = 0. This factors into (3x - 5)(x - 3) = 0, yielding x = 5/3 and x = 3. Factoring equation II, 6y^2 - 5y + 1 = 0, gives 6y^2 - 3y - 2y + 1 = 0, which factors into (3y - 1)(2y - 1) = 0, yielding y = 1/3 and y = 1/2. Since the smallest root of x (5/3) is larger than the largest root of y (1/2), x > y.

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