If x be real, then the minimum value of `x^(2)-8x+17` is

Options

A. `-1`
B. 0
C. 1
D. 2

Answer

To find the minimum value of the quadratic expression \( x^2 - 8x + 17 \), we can follow these steps:

### Step 1: Identify the quadratic expression
We start with the expression:
\[ f(x) = x^2 - 8x + 17 \]

### Step 2: Complete the square
To find the minimum value, we can complete the square for the quadratic expression. The general form of completing the square is:
\[ ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 + k \]
where \( k \) is a constant.

For our expression:
- \( a = 1 \)
- \( b = -8 \)
- \( c = 17 \)

We take half of \( b \) (which is -8), square it, and adjust the expression:
\[ f(x) = x^2 - 8x + 17 = (x^2 - 8x + 16) + 1 = (x - 4)^2 + 1 \]

### Step 3: Analyze the completed square
The expression \( (x - 4)^2 \) is always non-negative (i.e., \( \geq 0 \)) since it is a square term. The minimum value of \( (x - 4)^2 \) occurs when \( x - 4 = 0 \) or \( x = 4 \).

### Step 4: Find the minimum value
Substituting \( x = 4 \) back into the completed square expression:
\[ f(4) = (4 - 4)^2 + 1 = 0 + 1 = 1 \]

Thus, the minimum value of the expression \( x^2 - 8x + 17 \) is:
\[ \boxed{1} \]