factor 1/2(x^2 + 4x - 2) by completing the square 

To factor \(\frac{1}{2}(x^2 + 4x - 2)\) by completing the square, follow these steps:

### Step 1: Factor out \(\frac{1}{2}\)
We start by factoring out \(\frac{1}{2}\) from the entire expression:

\[
\frac{1}{2}(x^2 + 4x - 2) = \frac{1}{2} \left(x^2 + 4x - 2\right)
\]

### Step 2: Focus on the quadratic part \(x^2 + 4x\)
We'll work on completing the square for the quadratic expression \(x^2 + 4x\). To do this, take the coefficient of \(x\) (which is 4), divide it by 2, and square it:

\[
\left(\frac{4}{2}\right)^2 = 2^2 = 4
\]

Now add and subtract 4 inside the parentheses to complete the square:

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

### Step 3: Substitute back into the original expression
Now replace \(x^2 + 4x\) with \((x + 2)^2 - 4\):

\[
\frac{1}{2} \left((x + 2)^2 - 4 - 2\right)
\]

### Step 4: Simplify the constants
Simplify the constants inside the parentheses:

\[
-4 - 2 = -6
\]

Thus, the expression becomes:

\[
\frac{1}{2} \left((x + 2)^2 - 6\right)
\]

### Step 5: Distribute the \(\frac{1}{2}\)
Distribute the \(\frac{1}{2}\) over the expression:

\[
\frac{1}{2} (x + 2)^2 - \frac{1}{2} \times 6 = \frac{1}{2}(x + 2)^2 - 3
\]

### Final Factored Form:
The expression in factored form is:

\[
\frac{1}{2}(x + 2)^2 - 3
\]

This is the factored form of \(\frac{1}{2}(x^2 + 4x - 2)\) by completing the square.

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Would you like a more detailed explanation on any of the steps? Here are some questions to explore:

1. What is completing the square, and how is it useful?
2. How can I solve quadratic equations by completing the square?
3. Can completing the square always be used to factor quadratics?
4. How does the factor of \(\frac{1}{2}\) affect the process?
5. What are other ways to factor quadratic expressions?

**Tip:** Completing the square is a powerful technique not only for factoring but also for solving quadratic equations and finding the vertex of a parabola.

Factor 1/2(x^2 + 4x - 2) by completing the square.

Learn how to factor 1/2(x^2 + 4x - 2) by completing the square in this detailed step-by-step solution. This tutorial explains key concepts in algebra, such as factoring and completing the square, suitable for students in Grades 9-11.

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