The roots of the equation `4/(x^(2))=1+3/x` are

Options

A. `-1,-4`
B. `1,4`
C. `1,-4`
D. `-1,4`

Answer

To find the roots of the equation \( \frac{4}{x^2} = 1 + \frac{3}{x} \), we can follow these steps:

### Step-by-Step Solution:

1. **Rewrite the Equation**:
Start with the given equation:
\[ \frac{4}{x^2} = 1 + \frac{3}{x} \]

2. **Multiply through by \( x^2 \)**:
To eliminate the denominators, multiply both sides by \( x^2 \) (assuming \( x \neq 0 \)):
\[ 4 = x^2 + 3x \]

3. **Rearrange the Equation**:
Rearranging gives us a standard quadratic equation:
\[ x^2 + 3x - 4 = 0 \]

4. **Factor the Quadratic**:
We need to factor the quadratic equation. We are looking for two numbers that multiply to \(-4\) (the constant term) and add to \(3\) (the coefficient of \(x\)). The factors are \(4\) and \(-1\):
\[ (x + 4)(x - 1) = 0 \]

5. **Set Each Factor to Zero**:
Now, set each factor equal to zero:
\[ x + 4 = 0 \quad \text{or} \quad x - 1 = 0 \]

6. **Solve for \(x\)**:
Solving these equations gives:
\[ x = -4 \quad \text{and} \quad x = 1 \]

### Final Answer:
The roots of the equation are \( x = -4 \) and \( x = 1 \).

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