Solve each of the following equatins :
`x+(1)/(x)=2.5`

Answer

To solve the equation \( x + \frac{1}{x} = 2.5 \), we will follow these steps:

### Step 1: Eliminate the fraction
Multiply both sides of the equation by \( x \) (assuming \( x \neq 0 \)):
\[ x^2 + 1 = 2.5x \]

### Step 2: Rearrange the equation
Rearranging the equation gives us:
\[ x^2 - 2.5x + 1 = 0 \]

### Step 3: Convert to a standard form
To make calculations easier, we can multiply the entire equation by 2 to eliminate the decimal:
\[ 2x^2 - 5x + 2 = 0 \]

### Step 4: Factor the quadratic equation
We need to factor the quadratic equation \( 2x^2 - 5x + 2 = 0 \). We look for two numbers that multiply to \( 2 \times 2 = 4 \) and add to \( -5 \). The numbers are \( -4 \) and \( -1 \):
\[ 2x^2 - 4x - x + 2 = 0 \]
Now, we can group the terms:
\[ (2x^2 - 4x) + (-x + 2) = 0 \]
Factoring by grouping:
\[ 2x(x - 2) - 1(x - 2) = 0 \]
This gives us:
\[ (2x - 1)(x - 2) = 0 \]

### Step 5: Solve for \( x \)
Setting each factor to zero gives us the solutions:
1. \( 2x - 1 = 0 \) leads to \( x = \frac{1}{2} \)
2. \( x - 2 = 0 \) leads to \( x = 2 \)

### Final Solutions
Thus, the solutions to the equation \( x + \frac{1}{x} = 2.5 \) are:
\[ x = 2 \quad \text{and} \quad x = \frac{1}{2} \]

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