If `i^(2)=-1, 4+i`, and `4-i` are roots of which of the following equations ?

Options

A. `x^(2)+8x-17=0`
B. `x^(2)-8x+17=0`
C. `x^(2)-8x-17=0`
D. `x^(2)+10x-8=0`

Explanation

To find the equation for which \( 4+i \) and \( 4-i \) are roots, we can follow these steps:

### Step 1: Write the factors based on the roots
Since \( 4+i \) and \( 4-i \) are roots of the equation, we can express the polynomial as:
\[ (x - (4+i))(x - (4-i)) \]

### Step 2: Simplify the factors
Now, let's simplify the expression:
\[ (x - (4+i))(x - (4-i)) = (x - 4 - i)(x - 4 + i) \]
This can be recognized as a difference of squares.

### Step 3: Apply the difference of squares formula
Using the difference of squares formula \( (a-b)(a+b) = a^2 - b^2 \), where \( a = x - 4 \) and \( b = i \):
\[ = (x - 4)^2 - i^2 \]

### Step 4: Substitute \( i^2 \)
Since \( i^2 = -1 \), we can substitute:
\[ = (x - 4)^2 - (-1) = (x - 4)^2 + 1 \]

### Step 5: Expand the square
Now, we expand \( (x - 4)^2 \):
\[ = x^2 - 8x + 16 + 1 = x^2 - 8x + 17 \]

### Step 6: Write the final equation
Thus, we have:
\[ x^2 - 8x + 17 = 0 \]

### Conclusion
The equation for which \( 4+i \) and \( 4-i \) are roots is:
\[ x^2 - 8x + 17 = 0 \]