Multiple choice

If the roots of the equation $\displaystyle 3x^{2}+9x+2=0$ are in the ratio $m : n$, then find $\displaystyle \sqrt{\frac{m}{n}}+\sqrt{\frac{n}{m}}$

  1. $\displaystyle \frac{3\sqrt{3}}{\sqrt{2}}$
  2. $\displaystyle 3\sqrt{2}$
  3. $\displaystyle\frac{\sqrt{2}}{\sqrt{3}}$
  4. $\displaystyle 2\sqrt{3}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For 3x^2 + 9x + 2 = 0, m+n = -9/3 = -3 and mn = 2/3. We need sqrt(m/n) + sqrt(n/m) = (m+n)/sqrt(mn) = -3 / sqrt(2/3) = -3 * sqrt(3/2) = -3 * sqrt(3) / sqrt(2). The provided option A is positive, suggesting a sign error in the question or options.

AI explanation

Let the roots be mk and nk for some constant k, making their sum k(m+n) = -9/3 = -3 and their product k^2(mn) = 2/3. The required expression sqrt(m/n) + sqrt(n/m) can be rewritten with a common denominator as (m+n)/sqrt(mn). Squaring this expression gives (m+n)^2 / (mn). Substituting the sum and product relationships, we find the squared value equals (-3/k)^2 / (2/(3k^2)) = (9/k^2) * (3k^2/2) = 27/2. Taking the square root gives the final result of 3*sqrt(3)/sqrt(2).

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