If x^4 = x^2 + 1, where x > 0, then what is 2x^4 equal to?

Explanation

Let x^2 = t. The equation becomes t^2 - t - 1 = 0, giving t = \frac{1 \pm \sqrt{5}}{2}. Since x > 0, x^2 = \frac{1 + \sqrt{5}}{2}. Then x^4 = x^2 + 1 = \frac{1 + \sqrt{5}}{2} + 1 = \frac{3 + \sqrt{5}}{2}. Therefore, 2x^4 = 3 + \sqrt{5}.

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