X/16-X=1/X

To solve the equation \( \frac{x}{16} - x = \frac{1}{x} \), we can start by eliminating the fractions through multiplication.

1. First, let's identify a common denominator, which is \( 16x \): \[ 16x\left(\frac{x}{16} - x\right) = 16x\left(\frac{1}{x}\right) \]

2. Distributing this gives: \[ x^2 - 16x^2 = 16 \]

3. Combine the terms on the left: \[ -15x^2 = 16 \]

4. Now, divide by \(-15\): \[ x^2 = -\frac{16}{15} \]

5. This equation suggests that \( x^2 \) is negative, which means there are no real solutions because the square of a real number cannot be negative.

6. However, we can express the solutions in terms of complex numbers: \[ x = \pm \sqrt{-\frac{16}{15}} = \pm i\sqrt{\frac{16}{15}} = \pm \frac{4i}{\sqrt{15}} \]

Thus, the solutions to the equation \( \frac{x}{16} - x = \frac{1}{x} \) are: \[ x = \frac{4i}{\sqrt{15}} \quad \text{and} \quad x = -\frac{4i}{\sqrt{15}} \]