Multiple choice

If $\alpha, \beta$ are the roots of equation $4x^2 - \sqrt{13}x - 7 = 0$, then the value of $\alpha - \beta$ is

  1. $\displaystyle \frac{5 \sqrt{5} }{4}$
  2. $\displaystyle \frac{5}{6}$
  3. $\displaystyle \frac{\sqrt{5}}{6}$
  4. $2 \sqrt{5}$
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A Correct answer
Explanation

For 4x^2 - sqrt(13)x - 7 = 0, sum of roots alpha + beta = sqrt(13)/4, product alpha * beta = -7/4. (alpha - beta)^2 = (alpha + beta)^2 - 4*alpha*beta = (13/16) - 4(-7/4) = 13/16 + 7 = 125/16. alpha - beta = sqrt(125)/4 = 5*sqrt(5)/4.

AI explanation

Using the identity for the difference of roots, alpha - beta equals the square root of ((alpha + beta)^2 - 4*alpha*beta). From the equation 4x^2 - sqrt(13)x - 7 = 0, Vieta's formulas give the sum alpha + beta = sqrt(13)/4 and the product alpha*beta = -7/4. Substituting these gives the difference as the square root of (13/16 - 4*(-7/4)), which is the square root of 125/16. Simplifying this yields 5*sqrt(5)/4.

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