Find the roots of the following equations :
`x - (1)/(x) = 3, x != 0`

Answer

To find the roots of the equation \( x - \frac{1}{x} = 3 \) where \( x \neq 0 \), we can follow these steps:

### Step 1: Eliminate the fraction
Multiply both sides of the equation by \( x \) (since \( x \neq 0 \)):
\[ x^2 - 1 = 3x \]

### Step 2: Rearrange the equation
Rearranging the equation gives:
\[ x^2 - 3x - 1 = 0 \]

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

### Step 4: Apply the quadratic formula
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} \]

### Step 5: Simplify the expression
Calculating the discriminant:
\[ x = \frac{3 \pm \sqrt{9 + 4}}{2} \]
\[ x = \frac{3 \pm \sqrt{13}}{2} \]

### Step 6: Write the final roots
Thus, the roots of the equation are:
\[ x = \frac{3 + \sqrt{13}}{2} \quad \text{and} \quad x = \frac{3 - \sqrt{13}}{2} \]