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?

---

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!

Learn how to solve the rational equation 2/x² - 1 - 1/x = 1/2 step by step. This solution involves simplifying the equation, eliminating fractions, and solving a quadratic equation using the quadratic formula. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation