Multiple choice

If the equation $\displaystyle x+\frac{1}{x}=t$ has two real and distinct roots, then

  1. $-2 < t < 2$
  2. $-1 < t < 1$
  3. $t < -2$ or $t > 2$

  4. none of these
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C Correct answer
Explanation

x + 1/x = t. For x to be real, |t| >= 2. For roots to be distinct, |t| > 2. This means t > 2 or t < -2.

AI explanation

Multiply the equation x plus 1 over x equals t by x to get x squared minus tx plus 1 equals 0. For this quadratic equation to have two real and distinct roots, its discriminant must be strictly positive. The discriminant is t squared minus 4, so t squared minus 4 is greater than 0. This gives t is less than negative 2 or t is greater than 2.

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