how do i get this into vertex form: f(x)=x^2+8x+15

To convert the quadratic function \( f(x) = x^2 + 8x + 15 \) into vertex form, we need to complete the square.

Here are the steps:

### Step 1: Start with the given quadratic function
\[
f(x) = x^2 + 8x + 15
\]

### Step 2: Focus on the \(x^2 + 8x\) part
Leave the constant \(+15\) for now, and just work on completing the square for \(x^2 + 8x\).

\[
f(x) = (x^2 + 8x) + 15
\]

### Step 3: Complete the square
To complete the square, take half of the coefficient of \(x\) (which is 8), square it, and then add and subtract that value inside the function.

Half of 8 is 4, and \(4^2 = 16\). So, add and subtract 16 inside the parentheses:

\[
f(x) = (x^2 + 8x + 16 - 16) + 15
\]

\[
f(x) = ((x + 4)^2 - 16) + 15
\]

### Step 4: Simplify the expression
Now simplify by combining the constants \( -16 + 15 \).

\[
f(x) = (x + 4)^2 - 1
\]

### Final Answer: 
The vertex form of the function is:
\[
f(x) = (x + 4)^2 - 1
\]

In this form, the vertex is \( (-4, -1) \).

Do you have any questions or would you like further details?

Here are 5 related questions:
1. How do you find the vertex from the standard form of a quadratic?
2. How do you convert from vertex form back to standard form?
3. Can you complete the square if the coefficient of \(x^2\) is not 1?
4. How do you find the axis of symmetry for a quadratic function?
5. What is the relationship between the vertex form and transformations of a parabola?

**Tip:** Always check if the quadratic term has a coefficient of 1 before completing the square. If not, factor it out first!

Learn how to convert the quadratic function f(x) = x^2 + 8x + 15 into vertex form by completing the square. This step-by-step solution simplifies the quadratic equation, providing the vertex and related concepts for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation