convert the following quadratic equation to vertex form y=-x^2-16x-65

To convert the quadratic equation \( y = -x^2 - 16x - 65 \) into vertex form, we follow these steps:

### Step 1: Start with the standard form
The standard form of a quadratic equation is:

\[
y = ax^2 + bx + c
\]

For your equation:

\[
y = -x^2 - 16x - 65
\]

Here, \( a = -1 \), \( b = -16 \), and \( c = -65 \).

### Step 2: Complete the square
To convert the equation to vertex form, we need to complete the square:

1. Factor out the coefficient of \( x^2 \) (which is -1) from the \( x \)-terms:

\[
y = -\left(x^2 + 16x\right) - 65
\]

2. To complete the square, take half of the coefficient of \( x \) (which is 16), square it, and then add and subtract this value inside the parentheses:

Half of 16 is 8, and \( 8^2 = 64 \).

\[
y = -\left(x^2 + 16x + 64 - 64\right) - 65
\]

3. Simplify by factoring the perfect square trinomial and adjusting the constant:

\[
y = -\left((x + 8)^2 - 64\right) - 65
\]

4. Distribute the -1 and combine like terms:

\[
y = -(x + 8)^2 + 64 - 65
\]

\[
y = -(x + 8)^2 - 1
\]

### Step 3: Write the equation in vertex form
The vertex form of a quadratic equation is:

\[
y = a(x - h)^2 + k
\]

For this equation:

\[
y = -(x + 8)^2 - 1
\]

Here, \( h = -8 \) and \( k = -1 \). So, the vertex is at \( (-8, -1) \).

### Final Answer
The vertex form of the quadratic equation is:

\[
y = -(x + 8)^2 - 1
\]

Would you like to explore any specific details about this conversion or ask further questions? Here are some related questions:

1. How can you determine the vertex from the vertex form?
2. What is the significance of the vertex in a quadratic function?
3. How do you find the axis of symmetry from the vertex form?
4. What does the negative sign before \( (x + 8)^2 \) indicate about the graph?
5. How would the graph of \( y = -(x + 8)^2 - 1 \) differ from \( y = (x + 8)^2 - 1 \)?
6. Can you explain how to convert a quadratic function from vertex form back to standard form?
7. How can you use the vertex form to find the maximum or minimum value of a quadratic function?
8. What role does completing the square play in converting a quadratic equation to vertex form?

**Tip:** In vertex form, the coefficient \( a \) determines the direction and width of the parabola, while \( (h, k) \) represents its vertex, the highest or lowest point on the graph.

Learn how to convert the quadratic equation y = -x^2 - 16x - 65 to vertex form step-by-step. This problem covers key algebraic concepts including quadratic equations, vertex form, and completing the square method. Suitable for high school students studying algebra.

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