Let's solve the equation step by step:

Given equation: $\frac{2}{x^2} - 1 - \frac{1}{x} - 1 = \frac{1}{2}$

Step 1: Simplify the equation

Combine like terms: $\frac{2}{x^2} - \frac{1}{x} - 2 = \frac{1}{2}$

Step 2: Eliminate the fractions

To eliminate the fractions, multiply the entire equation by $2x^2$ (the least common multiple of all the denominators): $2x^2 \left(\frac{2}{x^2} - \frac{1}{x} - 2\right) = 2x^2 \cdot \frac{1}{2}$

This simplifies to: $4 - 2x - 4x^2 = x^2$

Step 3: Move all terms to one side of the equation

$4 - 2x - 4x^2 - x^2 = 0$

Combine like terms: $-5x^2 - 2x + 4 = 0$

Step 4: Solve the quadratic equation

The equation can be written as: $5x^2 + 2x - 4 = 0$

To solve this quadratic equation, we use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 5$, $b = 2$, and $c = -4$.

Calculate the discriminant: $\Delta = b^2 - 4ac = 2^2 - 4(5)(-4) = 4 + 80 = 84$

So the solutions are: $x = \frac{-2 \pm \sqrt{84}}{10}$

Simplifying further: $x = \frac{-2 \pm 2\sqrt{21}}{10} = \frac{-1 \pm \sqrt{21}}{5}$

Thus, the solutions are: $x = \frac{-1 + \sqrt{21}}{5} \quad \text{or} \quad x = \frac{-1 - \sqrt{21}}{5}$

Final Answer:

The solutions to the equation are: $x = \frac{-1 + \sqrt{21}}{5}, \quad x = \frac{-1 - \sqrt{21}}{5}$

Would you like more details or have any questions about this solution?

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Here are 5 related questions:

1. How can you solve a quadratic equation by completing the square?
2. What does the discriminant tell you about the nature of the roots of a quadratic equation?
3. How would the solution change if the equation had a different constant term?
4. How do you simplify expressions with square roots?
5. What are the steps to graphing a quadratic equation?

Tip: Always check your solutions by substituting them back into the original equation to ensure they satisfy it!