Multiple choice

Solve the following equation. $3\, \left(x^{2}\, +\, \displaystyle \frac{1}{x^{2}}\right)\, -\, 4\, \left(x\, -\, \displaystyle \frac{1}{x}\right)\, -\, 6\, =\, 0$

  1. $x\, =\, 1,\, -1,\, \displaystyle \frac{-2\, \pm\, \sqrt{13}}{3}$
  2. $x\, =\, 1,\, -1,\, \displaystyle \frac{2\, \pm\, \sqrt{13}}{3}$
  3. $x\, =\, 1,\, -1,\, \displaystyle \frac{4\, \pm\, \sqrt{13}}{6}$

  4. None of these 
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let y = x - 1/x. Then x^2 + 1/x^2 = y^2 + 2. Equation: 3(y^2 + 2) - 4y - 6 = 0 => 3y^2 - 4y = 0. y(3y - 4) = 0. y=0 or y=4/3. If x - 1/x = 0, x^2 = 1, x = +/- 1. If x - 1/x = 4/3, 3x^2 - 4x - 3 = 0, x = (4 +/- sqrt(16 + 36))/6 = (4 +/- sqrt(52))/6 = (4 +/- 2*sqrt(13))/6 = (2 +/- sqrt(13))/3.

AI explanation

Assuming the question intended to factor a 7 instead of a 6 from the constant, we substitute y = x - 1/x. Using the identity x^2 + 1/x^2 = y^2 + 2, the equation becomes 3(y^2 + 2) - 4y - 6 = 0. This simplifies to 3y^2 - 4y = 0, yielding y = 0 and y = 4/3. For y = 0, x = 1 or x = -1. For y = 4/3, solving x - 1/x = 4/3 gives 3x^2 - 4x - 3 = 0. Using the quadratic formula yields x = (4 +/- sqrt(52)) / 6, which simplifies to (2 +/- sqrt(13)) / 3.

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