`x^2-8x+5`
Which of the following is equivalent to the expression above?

Options

A. `(x-4)^2-11`
B. `(x-4)^2+11`
C. `(x+4)^2-11`
D. `(x+4)^2+11`

Answer

To solve the expression \( x^2 - 8x + 5 \) and rewrite it in the form \( (x - a)^2 + b \), we can follow these steps:

### Step 1: Identify the quadratic expression
The given expression is:
\[ x^2 - 8x + 5 \]

### Step 2: Complete the square
To complete the square, we need to focus on the quadratic and linear terms, which are \( x^2 - 8x \).

### Step 3: Find the value to complete the square
Take the coefficient of \( x \) (which is -8), divide it by 2, and square it:
\[ \left(\frac{-8}{2}\right)^2 = (-4)^2 = 16 \]

### Step 4: Rewrite the expression
Now, we can rewrite \( x^2 - 8x \) by adding and subtracting 16:
\[ x^2 - 8x = (x^2 - 8x + 16) - 16 = (x - 4)^2 - 16 \]

### Step 5: Incorporate the constant term
Now, we substitute this back into the original expression:
\[ x^2 - 8x + 5 = (x - 4)^2 - 16 + 5 \]
\[ = (x - 4)^2 - 11 \]

### Final Expression
Thus, the expression \( x^2 - 8x + 5 \) can be rewritten as:
\[ (x - 4)^2 - 11 \]

### Conclusion
The equivalent expression is:
\[ (x - 4)^2 - 11 \]