Multiple choice

If roots of the quadratic equation $3 a x ^ { 2 } + 2 b x + c = 0$ are in the ratio $2 : 3$, then which of the following statement is true?

  1. $8 a c = 25 b$
  2. $8 a c = 9 b ^ { 2 }$
  3. $8 b ^ { 2 } = 9 a c$
  4. $8 b ^ { 2 } = 25 a c$
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D Correct answer
Explanation

Let the roots be 2k and 3k. Sum of roots = 5k = -2b/(3a), so k = -2b/(15a). Product of roots = 6k^2 = c/(3a). Substituting k gives 6 * (4b^2 / 225a^2) = c/(3a), which simplifies to 8b^2 = 25ac.

AI explanation

Let the roots of 3ax squared plus 2bx plus c equals 0 be 2 lambda and 3 lambda. By Vieta's formulas, the sum of roots gives 2 lambda plus 3 lambda equals negative 2b divided by 3a, so lambda equals negative 2b divided by 15a. The product of roots gives 2 lambda multiplied by 3 lambda equals c divided by 3a, meaning 6 lambda squared equals c divided by 3a, or lambda squared equals c divided by 18a. Substituting the value of lambda into the product equation gives negative 2b divided by 15a all squared equal to c divided by 18a, so 4b squared divided by 225a squared equals c divided by 18a. Cross-multiplying yields 72b squared equals 225ac, which simplifies by dividing both sides by 9 to give 8b squared equals 25ac.

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