Multiple choice

If $\alpha$ and $\beta$ are the roots of the equation $3x^2 - 2x - 8 = 0$, find the value of $\alpha^2 - \alpha \beta + \beta^2$.

  1. $\dfrac{76}{9}$
  2. $\dfrac{25}{3}$
  3. $\dfrac{16}{3}$
  4. $\dfrac{32}{3}$
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A Correct answer
Explanation

For 3x^2 - 2x - 8 = 0, alpha + beta = 2/3 and alpha*beta = -8/3. The expression alpha^2 - alpha*beta + beta^2 is (alpha + beta)^2 - 3*alpha*beta = (2/3)^2 - 3*(-8/3) = 4/9 + 8 = 76/9.

AI explanation

Using Vieta's formulas for 3x^2 - 2x - 8 = 0, the sum of the roots (alpha + beta) is -(-2)/3 = 2/3, and the product (alpha * beta) is -8/3. The expression alpha^2 - alpha*beta + beta^2 can be rewritten using the identity (alpha + beta)^2 - 3*alpha*beta. Substituting the known values gives (2/3)^2 - 3(-8/3), which simplifies to 4/9 + 8 = 76/9.

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