Multiple choice

Directions: In the following item, Quantity I and Quantity II are given. Determine the relationship between the quantities and choose the appropriate option. In the quadratic equation ax2 + bx + c = 0 with roots α and β, where sum of roots is 10 and α2 + β2 = 34, Quantity I: The value of c/a Quantity II: 36

  1. Quantity I > Quantity II

  2. Quantity I < Quantity II

  3. Quantity I ≥ Quantity II

  4. Quantity I ≤ Quantity II

  5. Quantity I = Quantity II, or No relation
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Sum of roots alpha + beta = -b/a = 10. alpha^2 + beta^2 = (alpha+beta)^2 - 2*alpha*beta = 34. 100 - 2*alpha*beta = 34, so 2*alpha*beta = 66, alpha*beta = 33. Since alpha*beta = c/a, Quantity I = 33. Quantity II = 36. Thus, 33 < 36.

AI explanation

Let the roots be a and b. We are given the sum a plus b equals 10, and the sum of their squares a squared plus b squared equals 34. We know that a squared plus b squared equals the quantity a plus b squared minus 2ab, so 34 equals 100 minus 2ab, which means the product ab equals 33. For a quadratic equation, the ratio c over a represents the product of the roots, so Quantity I is 33. Since 33 is less than 36, Quantity I is less than Quantity II.

Keep practicing — related questions