Multiple choice

Solve the following equations: $\dfrac {x + \sqrt {x^{2} - 1}}{x - \sqrt {x^{2} - 1}} + \dfrac {x - \sqrt {x^{2} - 1}}{x + \sqrt {x^{2} - 1}} = 98$

  1. $\pm 1$
  2. $\pm5$
  3. $\pm 10$
  4. $\pm 4$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let y = (x + sqrt(x^2-1)) / (x - sqrt(x^2-1)). Then the equation is y + 1/y = 98. Solving y^2 - 98y + 1 = 0 gives y = 49 +/- sqrt(2400) = 49 +/- 20*sqrt(6). Alternatively, note that (x + sqrt(x^2-1))^2 = x^2 + x^2 - 1 + 2x*sqrt(x^2-1) = 2x^2 - 1 + 2x*sqrt(x^2-1). Testing x=5: (5+sqrt(24))/(5-sqrt(24)) = (5+sqrt(24))^2 = 25+24+10*sqrt(24) = 49+20*sqrt(6). The sum is 98.

AI explanation

Let the given expression be simplified by noting that (x + sqrt(x^2 - 1))(x - sqrt(x^2 - 1)) = x^2 - (x^2 - 1) = 1. Dividing the equation by the denominator of the first term or simplifying using identities leads to 2(x^2 + (x^2 - 1)) = 98, which simplifies to 4x^2 - 2 = 98. Solving gives 4x^2 = 100, so x^2 = 25, meaning x = ±5.

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