Multiple choice

If $\alpha$, $\beta$ are roots of the equation $ax^2 - bx - c = 0$, then $ \alpha^2 - \alpha \beta + \beta^2$ is equal to

  1. $\displaystyle \frac{b^2 + 3ac}{a^2}$
  2. $\displaystyle \frac{b^2 - 3ac}{a^2}$
  3. $\displaystyle \frac{b^2 + 2ac}{a^2}$
  4. $\displaystyle \frac{b^2 - 2ac}{a^2}$
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A Correct answer
Explanation

alpha + beta = b/a, alpha * beta = -c/a. alpha^2 - alpha*beta + beta^2 = (alpha + beta)^2 - 3*alpha*beta = (b/a)^2 - 3(-c/a) = b^2/a^2 + 3c/a = (b^2 + 3ac) / a^2.

AI explanation

Using Vieta's formulas for ax^2 - bx - c = 0, the sum of the roots is alpha + beta = b/a and the product is alpha * beta = -c/a. Rewrite the expression alpha^2 - alpha*beta + beta^2 as (alpha + beta)^2 - 3(alpha*beta). Substituting the known values gives (b/a)^2 - 3(-c/a), which simplifies to b^2/a^2 + 3ac/a^2. Combining these terms results in (b^2 + 3ac) / a^2.

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