The two roots of a quadratic equation are given as `x =(1)/(7)` and `x = (-1)/(8)` The equation can be written as:

Options

A. (7x-1)(8x - 1) = 0
B. (7x-1)(8x + 1) = 0
C. (7x + 1)(8x - 1) = 0
D. (7x + 1)(8x + 1) = 0

Answer

To derive the quadratic equation from the given roots \( x = \frac{1}{7} \) and \( x = -\frac{1}{8} \), we can follow these steps:

### Step 1: Identify the roots
The roots of the quadratic equation are given as:
- \( r_1 = \frac{1}{7} \)
- \( r_2 = -\frac{1}{8} \)

### Step 2: Calculate the sum of the roots
The sum of the roots \( S \) can be calculated using the formula:
\[ S = r_1 + r_2 = \frac{1}{7} + \left(-\frac{1}{8}\right) \]
To add these fractions, we need a common denominator, which is 56:
\[ S = \frac{8}{56} - \frac{7}{56} = \frac{1}{56} \]

### Step 3: Calculate the product of the roots
The product of the roots \( P \) is given by:
\[ P = r_1 \times r_2 = \frac{1}{7} \times -\frac{1}{8} = -\frac{1}{56} \]

### Step 4: Write the quadratic equation
Using the standard form of a quadratic equation based on the roots:
\[ x^2 - Sx + P = 0 \]
Substituting the values of \( S \) and \( P \):
\[ x^2 - \frac{1}{56}x - \frac{1}{56} = 0 \]

### Step 5: Eliminate the fractions
To eliminate the fractions, multiply the entire equation by 56:
\[ 56x^2 - x - 1 = 0 \]

### Step 6: Factor the quadratic equation
Now, we can factor the quadratic equation:
\[ 56x^2 - 8x + 7x - 1 = 0 \]
Grouping the terms:
\[ (56x^2 - 8x) + (7x - 1) = 0 \]
Factoring out common terms:
\[ 8x(7x - 1) + 1(7x - 1) = 0 \]
This gives us:
\[ (7x - 1)(8x + 1) = 0 \]

### Final Result
The quadratic equation can be written in factorized form as:
\[ (7x - 1)(8x + 1) = 0 \]