Multiple choice

Directions: There are two equations given in the problem. Solve the equations and answer accordingly. I. 7x2 - 25x - 12 = 0 II. 3y2 + 10y + 8 = 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: 7x^2 - 25x - 12 = 0 => 7x^2 - 28x + 3x - 12 = 0 => 7x(x-4) + 3(x-4) = 0 => (7x+3)(x-4) = 0. x = 4, -3/7. II: 3y^2 + 10y + 8 = 0 => 3y^2 + 6y + 4y + 8 = 0 => 3y(y+2) + 4(y+2) = 0 => (3y+4)(y+2) = 0. y = -4/3, -2. Comparing: 4 > -4/3, 4 > -2, -3/7 > -4/3, -3/7 > -2. Thus x > y.

AI explanation

Factoring equation I by splitting the middle term gives 7x^2 - 28x + 3x - 12 = 0, which becomes (7x + 3)(x - 4) = 0, yielding x = -3/7 and x = 4. Factoring equation II gives (3y + 4)(y + 2) = 0, yielding y = -4/3 and y = -2. Because both values of x are larger than both values of y, x is greater than y.

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