The roots of the quadratic equation `(x^(2)-8)/(x^(2) + 20) = (1)/(2)` are

Options

A. `pm 3`
B. `pm 2`
C. `pm 6`
D. `pm 4`

Answer

To find the roots of the quadratic equation \(\frac{x^2 - 8}{x^2 + 20} = \frac{1}{2}\), we will follow these steps:

### Step 1: Clear the fraction
Multiply both sides of the equation by \(2(x^2 + 20)\) to eliminate the fraction:
\[ 2(x^2 - 8) = 1(x^2 + 20) \]

### Step 2: Expand both sides
Now, expand both sides of the equation:
\[ 2x^2 - 16 = x^2 + 20 \]

### Step 3: Rearrange the equation
Bring all terms to one side of the equation:
\[ 2x^2 - x^2 - 16 - 20 = 0 \]
This simplifies to:
\[ x^2 - 36 = 0 \]

### Step 4: Factor the equation
Now, we can factor the equation:
\[ (x - 6)(x + 6) = 0 \]

### Step 5: Solve for \(x\)
Set each factor equal to zero:
1. \(x - 6 = 0 \Rightarrow x = 6\)
2. \(x + 6 = 0 \Rightarrow x = -6\)

### Conclusion
The roots of the quadratic equation are:
\[ x = 6 \quad \text{and} \quad x = -6 \]

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