`y^(2)-8y+2`
Which of the following is equivalent to the expression above?

Options

A. `(y-4)^(2)-14`
B. `(y-4)^(2)+14`
C. `(y+4)^(2)-14`
D. `(y+4)^(2)+14`

Answer

To solve the expression \( y^2 - 8y + 2 \) and rewrite it in a different form, we can complete the square. Here’s a step-by-step solution:

### Step 1: Identify the quadratic expression
We start with the expression:
\[ y^2 - 8y + 2 \]

### Step 2: Complete the square for the quadratic part
To complete the square for \( y^2 - 8y \), we take the coefficient of \( y \), which is \(-8\), divide it by 2, and then square it:
\[ \left( \frac{-8}{2} \right)^2 = (-4)^2 = 16 \]

### Step 3: Rewrite the expression
Now we can rewrite \( y^2 - 8y \) by adding and subtracting 16:
\[ y^2 - 8y = (y^2 - 8y + 16) - 16 = (y - 4)^2 - 16 \]

### Step 4: Substitute back into the original expression
Now we substitute this back into the original expression:
\[ y^2 - 8y + 2 = (y - 4)^2 - 16 + 2 \]
\[ = (y - 4)^2 - 14 \]

### Conclusion
Thus, the expression \( y^2 - 8y + 2 \) can be rewritten as:
\[ (y - 4)^2 - 14 \]

### Final Answer
The equivalent expression is:
\[ \boxed{(y - 4)^2 - 14} \]