Question

Complete the square to re-write the quadratic function in vertex form:
y, equals, minus, 2, x, squared, plus, 12, x, minus, 24
y=−2x
2
+12x−24

Answer 1

To rewrite the quadratic function \( y = -2x^2 + 12x - 24 \) in vertex form by completing the square, we can follow these steps:

1. **Factor out the coefficient of the \( x^2 \) term** (which is \(-2\)) from the first two terms:
\[
y = -2(x^2 - 6x) - 24
\]

2. **Complete the square inside the parentheses**. To do this, take the coefficient of \( x \) (which is \(-6\)), halve it (which gives \(-3\)), and then square it (which gives \(9\)):
\[
y = -2(x^2 - 6x + 9 - 9) - 24
\]
Rearranging gives:
\[
y = -2((x - 3)^2 - 9) - 24
\]

3. **Distribute the \(-2\)**:
\[
y = -2(x - 3)^2 + 18 - 24
\]

4. **Simplify the constant terms**:
\[
y = -2(x - 3)^2 - 6
\]

Now, the quadratic function in vertex form is:
\[
y = -2(x - 3)^2 - 6
\]

The vertex of the function is at \((3, -6)\).