A number to greater than 58 times times its reciprocal by `3/4`, What the number?

Options

A. `-8`
B. 12
C. `-12`
D. 8

Answer

To solve the problem, we need to find a number \( x \) such that it is greater than 58 times its reciprocal by \( \frac{3}{4} \).

### Step-by-Step Solution:

1. **Set up the equation**:
We know that the number \( x \) is greater than \( 58 \times \frac{1}{x} \) by \( \frac{3}{4} \). This can be expressed mathematically as:
\[ x = 58 \cdot \frac{1}{x} + \frac{3}{4} \]

2. **Multiply through by \( x \)**:
To eliminate the fraction, multiply both sides by \( x \):
\[ x^2 = 58 + \frac{3}{4}x \]

3. **Rearrange the equation**:
Rearranging gives us a standard quadratic equation:
\[ x^2 - \frac{3}{4}x - 58 = 0 \]

4. **Clear the fraction**:
To make calculations easier, multiply the entire equation by 4 to eliminate the fraction:
\[ 4x^2 - 3x - 232 = 0 \]

5. **Use the quadratic formula**:
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 4 \), \( b = -3 \), and \( c = -232 \).

6. **Calculate the discriminant**:
First, calculate \( b^2 - 4ac \):
\[ (-3)^2 - 4 \cdot 4 \cdot (-232) = 9 + 3712 = 3721 \]

7. **Find the square root of the discriminant**:
\[ \sqrt{3721} = 61 \]

8. **Substitute back into the quadratic formula**:
Now substitute back into the formula:
\[ x = \frac{3 \pm 61}{8} \]

9. **Calculate the two possible values for \( x \)**:
- First solution:
\[ x = \frac{3 + 61}{8} = \frac{64}{8} = 8 \]
- Second solution:
\[ x = \frac{3 - 61}{8} = \frac{-58}{8} = -7.25 \]

10. **Select the valid solution**:
Since we are looking for a positive number, we take \( x = 8 \).

### Final Answer:
The number is \( \boxed{8} \).