Rewrite the function by completing the square.

Question

\[f(x)=x^{2}+12x-69\]
\[f(x)=(x+\]

\[)^2+\]

Answer 1

To rewrite the function \( f(x) = x^2 + 12x - 69 \) by completing the square, follow these steps:

1. Start with the quadratic part of the function, which is \( x^2 + 12x \).
2. To complete the square, take the coefficient of \( x \) (which is 12), divide it by 2, and square the result: \[ \left(\frac{12}{2}\right)^2 = 6^2 = 36 \]
3. Rewrite the quadratic expression by adding and subtracting this square: \[ x^2 + 12x = (x^2 + 12x + 36) - 36 = (x + 6)^2 - 36 \]
4. Substitute this back into \( f(x) \): \[ f(x) = (x + 6)^2 - 36 - 69 \]
5. Combine the constant terms: \[ -36 - 69 = -105 \]
6. Thus, the function can be rewritten as: \[ f(x) = (x + 6)^2 - 105 \]

So the final answer is: \[ f(x) = (x + 6)^2 - 105 \]