The two roots of the equation f(x)=f`((x+8)/(x-1))` are:

Options

A. 2,-2
B. 4,-2
C. `-4,-2`
D. 2,4

Answer

To solve the equation \( f(x) = f\left(\frac{x+8}{x-1}\right) \) and find its roots, we can follow these steps:

### Step 1: Set up the equation
We start with the equation:
\[ f(x) = f\left(\frac{x+8}{x-1}\right) \]
This implies that the two arguments must be equal or one must be the inverse of the other. We can express this as:
\[ x = \frac{x+8}{x-1} \]

### Step 2: Clear the fraction
To eliminate the fraction, we can multiply both sides by \( (x-1) \):
\[ x(x-1) = x + 8 \]

### Step 3: Expand and rearrange
Expanding the left side gives:
\[ x^2 - x = x + 8 \]
Now, we can rearrange the equation by moving all terms to one side:
\[ x^2 - x - x - 8 = 0 \]
This simplifies to:
\[ x^2 - 2x - 8 = 0 \]

### Step 4: Factor the quadratic equation
Next, we need to factor the quadratic equation \( x^2 - 2x - 8 = 0 \). We look for two numbers that multiply to \(-8\) and add up to \(-2\). These numbers are \(-4\) and \(2\):
\[ (x - 4)(x + 2) = 0 \]

### Step 5: Solve for the roots
Setting each factor equal to zero gives us the roots:
1. \( x - 4 = 0 \) leads to \( x = 4 \)
2. \( x + 2 = 0 \) leads to \( x = -2 \)

Thus, the two roots of the equation are:
\[ x = 4 \quad \text{and} \quad x = -2 \]

### Final Answer
The two roots of the equation are \( 4 \) and \( -2 \).

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