Multiple choice

Find the roots of the following quadratic equations by using the quadratic formula $3a^2x^2 + 8abx + 4b^2 = 0, a \neq 0$

  1. $\displaystyle \frac{2b}{a}, \frac{-2b}{3a}$
  2. $\displaystyle \frac{-3b}{2a}, \frac{-2b}{3a}$
  3. $\displaystyle \frac{-2b}{a}, \frac{2b}{3a}$
  4. $\displaystyle \frac{-2b}{a}, \frac{-2b}{3a}$
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D Correct answer
Explanation

Using the quadratic formula x = (-b +/- sqrt(b^2 - 4ac)) / 2a, where A=3a^2, B=8ab, C=4b^2. The discriminant is (8ab)^2 - 4(3a^2)(4b^2) = 64a^2b^2 - 48a^2b^2 = 16a^2b^2. Roots are (-8ab +/- 4ab) / 6a^2, yielding -12ab/6a^2 = -2b/a and -4ab/6a^2 = -2b/3a.

AI explanation

Using the quadratic formula with A equal to 3a squared, B equal to 8ab and C equal to 4b squared, the discriminant is 64a squared b squared minus 48a squared b squared, which is 16a squared b squared. The square root of the discriminant is 4ab, giving the roots as negative 8ab plus or minus 4ab, all divided by 6a squared. This simplifies to the two roots negative 2b divided by a and negative 2b divided by 3a.

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