Multiple choice

Find the roots of the following equation: $\displaystyle\,2\left(x^2\,+\,\frac{1}{x^2}\right)\,-\,9\,\left(x\,+\,\frac{1}{x}\right)\,+\,14\,=\,0$, then

  1. $\displaystyle\,x\,=\,-1, \,2,\, \frac{1}{2}$
  2. $\displaystyle\,x\,=\,1, \,2,\, \frac{1}{2}$
  3. $\displaystyle\,x\,=\,1, \,-2,\, \frac{1}{2}$
  4. $\displaystyle\,x\,=\,1, \,2,\, \frac{1}{4}$
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B Correct answer
Explanation

Let y = x + 1/x. Then x^2 + 1/x^2 = y^2 - 2. Equation: 2(y^2 - 2) - 9y + 14 = 0. 2y^2 - 9y + 10 = 0. (2y - 5)(y - 2) = 0. y = 2 or y = 2.5. If x + 1/x = 2, x = 1. If x + 1/x = 2.5, x = 2 or 1/2.

AI explanation

Using the algebraic identity x^2 + 1/x^2 = (x + 1/x)^2 - 2, we substitute m = x + 1/x to rewrite the equation as 2(m^2 - 2) - 9m + 14 = 0. Simplifying this gives 2m^2 - 9m + 10 = 0, which factors into (2m - 5)(m - 2) = 0. Solving x + 1/x = 5/2 yields x = 2 and x = 1/2, while solving x + 1/x = 2 gives x = 1. The roots are 1, 2, and 1/2.

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