Multiple choice

The roots of the equation $x+\cfrac{1}{x}=3\cfrac{1}{3}, x\neq 0$, are

  1. $3,1$
  2. $3,\cfrac{1}{3}$
  3. $3,-\cfrac{1}{3}$
  4. $-3,-\cfrac{1}{3}$
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B Correct answer
Explanation

x + 1/x = 10/3. Multiply by 3x: 3x^2 + 3 = 10x. 3x^2 - 10x + 3 = 0. (3x - 1)(x - 3) = 0. Roots are 3 and 1/3.

AI explanation

Rewrite the mixed number 3 1/3 as the improper fraction 10/3, making the equation x + 1/x = 10/3. Multiplying the entire equation by 3x eliminates the fractions, resulting in the quadratic equation 3x^2 + 3 = 10x, or 3x^2 - 10x + 3 = 0. We factor this quadratic by finding two numbers that multiply to 9 and add to -10, yielding (3x - 1)(x - 3) = 0. The roots of the equation are 3 and 1/3.

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