It think your 3 step is wrong there won't be -6 it goes to 0 side

You're correct. Let's revisit the problem to avoid the mistake in Step 3 and solve it correctly.

Given equation:

\[
\frac{x + 2}{6} - \frac{1}{x + 2} - \frac{1}{6} = 0
\]

### Step 1: Simplify the equation

First, let's combine the fractions on the left-hand side of the equation:

\[
\frac{x + 2}{6} - \frac{1}{6} - \frac{1}{x + 2} = 0
\]

We can combine the first two terms since they have the same denominator:

\[
\frac{(x + 2) - 1}{6} - \frac{1}{x + 2} = 0
\]

This simplifies to:

\[
\frac{x + 1}{6} - \frac{1}{x + 2} = 0
\]

### Step 2: Eliminate the fractions

To eliminate the fractions, multiply the entire equation by \(6(x + 2)\) (the least common denominator):

\[
(x + 1)(x + 2) - 6 = 0
\]

### Step 3: Expand and simplify correctly

Now, let's expand and simplify the expression:

\[
(x + 1)(x + 2) = x^2 + 3x + 2
\]

So the equation becomes:

\[
x^2 + 3x + 2 - 6 = 0
\]

However, since \( -6 \) is subtracted on the same side as the quadratic terms, the corrected equation should be:

\[
x^2 + 3x + 2 = 6
\]

### Step 4: Move constant term to the other side

Next, move the constant term to the other side:

\[
x^2 + 3x + 2 - 6 = 0
\]

Simplifying further:

\[
x^2 + 3x - 4 = 0
\]

### Step 5: Complete the square

To complete the square:

1. Start with the quadratic equation:

\[
x^2 + 3x = 4
\]

2. Take half the coefficient of \(x\), square it, and add to both sides:

\[
\left(\frac{3}{2}\right)^2 = \frac{9}{4}
\]

Add \(\frac{9}{4}\) to both sides:

\[
x^2 + 3x + \frac{9}{4} = 4 + \frac{9}{4}
\]

3. Simplify the right side:

\[
x^2 + 3x + \frac{9}{4} = \frac{16}{4} + \frac{9}{4} = \frac{25}{4}
\]

4. The left side is now a perfect square:

\[
\left(x + \frac{3}{2}\right)^2 = \frac{25}{4}
\]

### Step 6: Solve for \(x\)

Take the square root of both sides:

\[
x + \frac{3}{2} = \pm \frac{5}{2}
\]

### Step 7: Find the solutions

Solve for \(x\):

\[
x = -\frac{3}{2} \pm \frac{5}{2}
\]

This gives two solutions:

\[
x = 1 \quad \text{or} \quad x = -4
\]

### Final Answer:

The solutions are:

\[
x = 1 \quad \text{or} \quad x = -4
\]

Thank you for pointing out the error! If you have any further questions or would like more details on any part of the solution, feel free to ask. Here are some related questions you might find helpful:

1. How can you verify the solutions of a quadratic equation?
2. What are the benefits of the completing the square method?
3. How do you apply the quadratic formula to solve any quadratic equation?
4. What is the discriminant, and how does it determine the nature of roots?
5. How can quadratic equations be used in real-life situations?

**Tip:** Completing the square is particularly useful for deriving the quadratic formula and finding the vertex form of a parabola.

Learn how to solve the equation (x + 2)/6 - 1/(x + 2) - 1/6 = 0 with a detailed step-by-step solution. This problem covers key algebraic concepts, including equations involving fractions and quadratic equations. Suitable for students in Grades 10-12.

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