Question

Complete the square to re-write the quadratic function in vertex form:
y, equals, minus, 9, x, squared, minus, 108, x, minus, 333
y=−9x
2
−108x−333

Answer 1

To rewrite the quadratic function \( y = -9x^2 - 108x - 333 \) in vertex form by completing the square, follow these steps:

1. **Factor out the coefficient of \( x^2 \)** from the first two terms:
\[
y = -9(x^2 + 12x) - 333
\]

2. **Complete the square** inside the parentheses. To do this, take the coefficient of \( x \) (which is \( 12 \)), divide it by 2 to get \( 6 \), and then square it to obtain \( 36 \).

3. **Add and subtract this square inside the parentheses**:
\[
y = -9(x^2 + 12x + 36 - 36) - 333
\]
Simplifying this gives:
\[
y = -9((x + 6)^2 - 36) - 333
\]

4. **Distribute the -9**:
\[
y = -9(x + 6)^2 + 324 - 333
\]
This simplifies to:
\[
y = -9(x + 6)^2 - 9
\]

Now the quadratic function in vertex form is:
\[
y = -9(x + 6)^2 - 9
\]

The vertex of the parabola represented by this equation is at \( (-6, -9) \).