Question

Rewrite the function by completing the square.
\[f(x)=x^{2}+12x-69\]
\[f(x)=(x+\]

\[)^2+\]

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:
\[
f(x) = (x + 6)^2 - 105
\]