Multiple choice

If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 4x + 1 = 0$, then the value of $\displaystyle \frac{1}{\alpha + 2 \beta} + \frac{1}{\beta + 2\alpha}$ is _________.

  1. $\displaystyle \frac{12}{13}$
  2. $\displaystyle \frac{6}{5}$
  3. $\displaystyle \frac{8}{15}$
  4. $\displaystyle \frac{12}{17}$
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D Correct answer
Explanation

Given roots alpha and beta of 2x^2 - 4x + 1 = 0, we have alpha + beta = 2 and alpha * beta = 0.5. The expression simplifies to (alpha + beta + 2alpha + 2beta) / ((alpha + 2beta)(beta + 2alpha)) = 3(alpha + beta) / (alpha*beta + 2alpha^2 + 2beta^2 + 4alpha*beta). Using symmetric sums, the result is 12/17.

AI explanation

From the equation 2x^2 - 4x + 1 = 0, the sum of the roots is alpha + beta = 2 and the product is alpha*beta = 1/2. The given expression simplifies by finding a common denominator as (beta + 2*alpha + alpha + 2*beta) divided by (alpha + 2*beta)(beta + 2*alpha), which equals 3(alpha + beta) divided by 2*alpha^2 + 5*alpha*beta + 2*beta^2. Substituting the values gives 6 divided by (2((alpha + beta)^2 - 2*alpha*beta) + 5*alpha*beta) = 6 divided by (2(4 - 1) + 5/2) = 6 divided by 8.5 = 12/17.

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