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: Multiply through by \( x \)
To eliminate the fraction, we can multiply both sides of the equation by \( x \) (assuming \( x \neq 0 \)):
\[ x^2 + 1 = 2.5x \]

### Step 2: Rearrange the equation
Next, we will rearrange the equation to set it to zero:
\[ x^2 - 2.5x + 1 = 0 \]

### Step 3: Identify coefficients
Now, we identify the coefficients \( a \), \( b \), and \( c \) from the quadratic equation \( ax^2 + bx + c = 0 \):
- \( a = 1 \)
- \( b = -2.5 \)
- \( c = 1 \)

### Step 4: Calculate the discriminant
The discriminant \( D \) is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values:
\[ D = (-2.5)^2 - 4 \cdot 1 \cdot 1 = 6.25 - 4 = 2.25 \]

### Step 5: Find the roots using the quadratic formula
The roots of the quadratic equation can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
Substituting the values of \( b \) and \( D \):
\[ x = \frac{-(-2.5) \pm \sqrt{2.25}}{2 \cdot 1} = \frac{2.5 \pm 1.5}{2} \]

### Step 6: Calculate the two possible values for \( x \)
Now we will calculate the two possible values for \( x \):
1. For \( x_1 \):
\[ x_1 = \frac{2.5 + 1.5}{2} = \frac{4}{2} = 2 \]
2. For \( x_2 \):
\[ x_2 = \frac{2.5 - 1.5}{2} = \frac{1}{2} = 0.5 \]

### Final Answer
The solutions to the equation \( x + \frac{1}{x} = 2.5 \) are:
\[ x = 2 \quad \text{and} \quad x = 0.5 \]

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