Use completing the square to rewrite the given function in vertex form. Determine whether the vertex is a maximum or minimum value. Select the TWO correct answers.

y = x^2 + 10x + 15
Responses
A	The vertex form is y = 
 + 10The vertex form is y = ( x + 5 ) 2 + 10
B	The vertex form is y = 
 + 10.The vertex form is y = ( x − 5 ) 2 + 10.
C	The vertex form is y = 
 − 10.The vertex form is y = ( x + 5 ) 2 − 10.
D	The vertex is a maximum value at point (5, 10)The vertex is a maximum value at point (5, 10)
E	The vertex is a minimum value at point (−5, −10)

To rewrite the given function \( y = x^2 + 10x + 15 \) in vertex form using the method of completing the square, we follow these steps:

### Step 1: Start with the quadratic expression
The given equation is:
\[
y = x^2 + 10x + 15
\]

### Step 2: Completing the square
We take the coefficient of \( x \), which is 10, divide it by 2, and then square it. This helps in creating a perfect square trinomial.
\[
\left( \frac{10}{2} \right)^2 = 25
\]

Now, add and subtract 25 inside the equation:
\[
y = (x^2 + 10x + 25) - 25 + 15
\]
\[
y = (x + 5)^2 - 10
\]

### Step 3: Vertex form
The equation is now in vertex form:
\[
y = (x + 5)^2 - 10
\]
So, the correct vertex form is \( y = (x + 5)^2 - 10 \).

### Step 4: Determine the vertex
The vertex form of a quadratic function is \( y = a(x - h)^2 + k \), where the vertex is at the point \( (h, k) \). In this case, the vertex is \( (-5, -10) \).

### Step 5: Maximum or Minimum
Since the coefficient of \( x^2 \) (which is 1) is positive, the parabola opens upwards, meaning the vertex represents a **minimum value** at \( (-5, -10) \).

### Correct Answers:
- **C**: The vertex form is \( y = (x + 5)^2 - 10 \).
- **E**: The vertex is a minimum value at point \( (-5, -10) \).

### Would you like more details or have any questions?
Here are some related questions for further practice:

1. How do you determine if the vertex is a maximum or minimum based on the coefficient of \(x^2\)?
2. What would the vertex form be if the quadratic was \( y = x^2 - 10x + 15 \)?
3. Can you explain why adding and subtracting the same value inside the equation helps in completing the square?
4. How does the sign of the number added in the perfect square trinomial affect the vertex form?
5. What is the general procedure for completing the square for any quadratic function?

**Tip:** Always remember that the coefficient of \(x^2\) tells you if the parabola opens upwards (minimum) or downwards (maximum).

Use completing the square to rewrite the given function in vertex form. Determine whether the vertex is a maximum or minimum value. Select the TWO correct answers. y = x^2 + 10x + 15

Learn how to rewrite the quadratic function y = x^2 + 10x + 15 in vertex form using the completing the square method. This problem explains how to identify the vertex as a minimum value and walks through the step-by-step process of solving quadratic equations, suitable for students in Grades 9-11.

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