Certainly! Let's revisit **Step 1** where we factored out the 0.5 from the quadratic equation. The original equation is:

\[
y = 0.5x^2 + 4x + 1
\]

### Understanding the Factor Out Process

When we "factor out" a number, we are essentially dividing each term by that number and placing it outside a set of parentheses. Here's the process step by step:

#### 1. Identify the terms to factor from:
In this equation, the first two terms are \(0.5x^2\) and \(4x\).

#### 2. Factor out the common factor:
Notice that both terms share a common factor with respect to their coefficients. Specifically, 0.5 is a factor that can be factored out of both \(0.5x^2\) and \(4x\). 

To factor 0.5 out:

- **Divide each term by 0.5**: 
  - \( 0.5x^2 \div 0.5 = x^2 \)
  - \( 4x \div 0.5 = 8x \)

Thus, factoring 0.5 out gives:

\[
y = 0.5(x^2 + 8x) + 1
\]

Here’s a breakdown of what happened:
- We took the 0.5 common to both terms \(0.5x^2\) and \(4x\), leaving behind the expression \(x^2 + 8x\) inside the parentheses.
- The constant term \(+1\) remains outside because it is not involved in the factoring process.

### Why is this Important?
Factoring out the 0.5 is a necessary step when you want to complete the square within the parentheses, as it simplifies the quadratic expression and allows for the equation to be rewritten in vertex form.

Would you like more details or have any further questions? Here are five related questions:

1. Why do we need to complete the square when converting to vertex form?
2. What is the geometric interpretation of factoring in a quadratic equation?
3. How does factoring out affect the graph of a quadratic function?
4. Can you factor out other values besides 0.5 in different equations?
5. What would happen if we didn't factor out 0.5 before completing the square?

**Tip:** Always look for common factors in quadratic equations before completing the square—it simplifies the process and helps avoid mistakes!

Explore the process of factoring out in quadratic equations with a detailed step-by-step explanation. Learn how to factor out common factors to simplify quadratic expressions and prepare them for completing the square. Suitable for high school students studying algebra.

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