Multiple choice

If 8d and 5d are the roots of the quadratic equation 6p2 + rp + s = 0, where d is an integer, then determine the possible value of (r2 + 3s).

  1. 10206

  2. 27216

  3. 45336

  4. 54436
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Roots are 8d and 5d. Sum = 13d = -r/6. Product = 40d^2 = s/6. r = -78d, s = 240d^2. r^2 + 3s = (-78d)^2 + 3(240d^2) = 6084d^2 + 720d^2 = 6804d^2. For d=2, 6804*4 = 27216.

AI explanation

Using the sum and product of roots formulas for 6p^2 + rp + s = 0, we have 8d + 5d = -r / 6, which simplifies to r = -78d, and (8d)(5d) = s / 6, which simplifies to s = 240d^2. Substituting these into the expression (r^2 + 3s) gives (-78d)^2 + 3(240d^2), which simplifies to 6084d^2 + 720d^2 and equals 6804d^2. Checking the options to find a perfect square when divided by 6804, we divide 27216 by 6804 to get 4, meaning d^2 is 4 and d is 2, confirming the value is 27216.

Keep practicing — related questions