Solve each of the following equatins :
`48x^(2)-13x-1=0`

Answer

To solve the quadratic equation \( 48x^2 - 13x - 1 = 0 \), we will follow these steps:

### Step 1: Identify coefficients
The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \). Here, we have:
- \( a = 48 \)
- \( b = -13 \)
- \( c = -1 \)

### Step 2: Calculate the discriminant
The discriminant \( D \) is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (-13)^2 - 4 \cdot 48 \cdot (-1) \]
Calculating \( D \):
\[ D = 169 + 192 = 361 \]

### Step 3: Apply the quadratic formula
The solutions for \( x \) can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
Substituting \( b = -13 \) and \( D = 361 \):
\[ x = \frac{-(-13) \pm \sqrt{361}}{2 \cdot 48} \]
This simplifies to:
\[ x = \frac{13 \pm 19}{96} \]

### Step 4: Calculate the two possible values for \( x \)
Now we will calculate the two possible values for \( x \):
1. First value:
\[ x_1 = \frac{13 + 19}{96} = \frac{32}{96} = \frac{1}{3} \]

2. Second value:
\[ x_2 = \frac{13 - 19}{96} = \frac{-6}{96} = \frac{-1}{16} \]

### Final Solutions
The solutions to the quadratic equation \( 48x^2 - 13x - 1 = 0 \) are:
\[ x = \frac{1}{3} \quad \text{and} \quad x = \frac{-1}{16} \]

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