Multiple choice

Find the roots of the following equation: $\displaystyle\,x^2\,+\,\frac{12}{x^2}\,=\,7$, then

  1. $x\,=\,\pm\,2,\,\pm\,\sqrt5$
  2. $x\,=\,\pm\,2,\,\pm\,\sqrt3$
  3. $x\,=\,\pm\,3,\,\pm\,\sqrt3$
  4. $x\,=\,\pm\,2,\,\pm\,\sqrt2$
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B Correct answer
Explanation

Let u = x^2. The equation is u + 12/u = 7, so u^2 - 7u + 12 = 0. Factoring gives (u-3)(u-4) = 0. So x^2 = 3 or x^2 = 4. Thus x = +/- sqrt(3) or x = +/- 2.

AI explanation

Multiplying the equation x^2 + 12/x^2 = 7 by x^2 gives the biquadratic equation x^4 - 7x^2 + 12 = 0. Factoring this quadratic in x^2 yields (x^2 - 3)(x^2 - 4) = 0. Equating each factor to zero gives x^2 = 3 or x^2 = 4, so the roots are x = plus or minus the square root of 3 and x = plus or minus 2.

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