Solve each of the following equatins :
`x=(3x+1)/(4x)`

Answer

To solve the equation \( x = \frac{3x + 1}{4x} \), we will follow these steps:

### Step 1: Eliminate the fraction
Multiply both sides of the equation by \( 4x \) to eliminate the fraction:

\[ 4x^2 = 3x + 1 \]

### Step 2: Rearrange the equation
Rearranging the equation gives us:

\[ 4x^2 - 3x - 1 = 0 \]

### Step 3: Identify coefficients
In the quadratic equation \( ax^2 + bx + c = 0 \), we identify the coefficients:
- \( a = 4 \)
- \( b = -3 \)
- \( c = -1 \)

### Step 4: Use 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 \( c \):

\[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 4 \cdot (-1)}}{2 \cdot 4} \]

### Step 5: Simplify the expression
Calculating the discriminant:

\[ x = \frac{3 \pm \sqrt{9 + 16}}{8} \]
\[ x = \frac{3 \pm \sqrt{25}}{8} \]

### Step 6: Calculate the square root
Now, we find the square root of 25:

\[ x = \frac{3 \pm 5}{8} \]

### Step 7: Find the two possible values of \( x \)
Calculating the two possible values:

1. \( x = \frac{3 + 5}{8} = \frac{8}{8} = 1 \)
2. \( x = \frac{3 - 5}{8} = \frac{-2}{8} = -\frac{1}{4} \)

### Final Answer
The solutions to the equation are:

\[ x = 1 \quad \text{and} \quad x = -\frac{1}{4} \]

---