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 \( f(x) = f\left(\frac{x+8}{x-1}\right) \), we will follow these steps:

### Step 1: Set up the equation
We start with the given equation:
\[ f(x) = f\left(\frac{x+8}{x-1}\right) \]
This implies that the two expressions must be equal for some values of \( x \).

### Step 2: Use the property of functions
Since \( f(a) = f(b) \) implies \( a = b \) for a one-to-one function, we can set:
\[ x = \frac{x+8}{x-1} \]
This will help us find the values of \( x \) that satisfy the equation.

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ x(x-1) = x + 8 \]
Expanding both sides:
\[ x^2 - x = x + 8 \]

### Step 4: Rearrange the equation
Rearranging the equation leads to:
\[ x^2 - 2x - 8 = 0 \]

### Step 5: Apply the quadratic formula
To find the roots of the quadratic equation \( x^2 - 2x - 8 = 0 \), we use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = -2, c = -8 \).

### Step 6: Calculate the discriminant
First, calculate the discriminant \( D \):
\[ D = b^2 - 4ac = (-2)^2 - 4 \cdot 1 \cdot (-8) = 4 + 32 = 36 \]

### Step 7: Substitute into the quadratic formula
Now substituting the values into the formula:
\[ x = \frac{2 \pm \sqrt{36}}{2 \cdot 1} = \frac{2 \pm 6}{2} \]

### Step 8: Solve for the roots
Calculating the two possible values:
1. \( x = \frac{2 + 6}{2} = \frac{8}{2} = 4 \)
2. \( x = \frac{2 - 6}{2} = \frac{-4}{2} = -2 \)

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

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