Multiple choice

Solve the following quadratic equations by factorization method: $x+\dfrac{1}{x}=\dfrac{26}{5}$

  1. $\left \{ 5, \dfrac{1}{5} \right \}$ 
  2. $\left \{ -5, \dfrac{1}{5} \right \}$ 
  3. $\left \{ 5, \dfrac{-1}{5} \right \}$ 

  4. None of these
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A Correct answer
Explanation

x + 1/x = 26/5. (x^2 + 1) / x = 26/5. 5x^2 - 26x + 5 = 0. 5x^2 - 25x - x + 5 = 0. 5x(x - 5) - 1(x - 5) = 0. (5x - 1)(x - 5) = 0. Roots are 5 and 1/5.

AI explanation

Multiply the equation x + 1/x = 26/5 by 5x to clear the fractions, yielding 5x^2 + 5 = 26x. Rearrange this into standard quadratic form: 5x^2 - 26x + 5 = 0. By splitting the middle term, we factor it as 5x^2 - 25x - x + 5 = 0, which gives (x - 5)(5x - 1) = 0. The roots of the equation are 5 and 1/5.

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