Solve it by completing squares method x+2/6 - 1/x+2 - 1/6=0

The equation to solve is:

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

Let's solve it step by step by completing the square.

### Step 1: Simplify the equation
First, let's combine the fractions on the left-hand side of the equation.

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

We can rewrite this as:

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

Now, combine the first two fractions:

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

Simplify the numerator of the first fraction:

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

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

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

### Step 3: Expand and simplify
Expand the expression:

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

So the equation becomes:

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

Simplify further:

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

### Step 4: Complete the square
To complete the square, we need to transform \(x^2 + 3x - 4 = 0\) into a perfect square.

1. Start by moving the constant term to the other side:

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

2. Take half of the coefficient of \(x\), square it, and add it 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. Now, the left side is a perfect square:

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

### Step 5: Solve for \(x\)
Take the square root of both sides:

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

### Step 6: Find the solutions
Solve for \(x\) by isolating \(x\):

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

This gives us two possible solutions:

\[
x = \frac{2}{2} = 1 \quad \text{or} \quad x = -\frac{8}{2} = -4
\]

### Final Answer:
The solutions are:

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

Would you like further details on any step or have additional questions? Here are some related questions you might find interesting:

1. How do you solve quadratic equations by the quadratic formula?
2. What are the conditions for a quadratic equation to have real roots?
3. Can completing the square method be used for all quadratic equations?
4. How can you verify the solutions to a quadratic equation?
5. What is the geometric interpretation of completing the square?

**Tip:** Always check your solutions by substituting them back into the original equation to ensure they satisfy the equation.

Learn how to solve the equation x + 2 / 6 - 1 / (x + 2) - 1 / 6 = 0 step-by-step using the completing the square method. This problem involves algebraic manipulation and solving equations involving fractions. Suitable for advanced high school students.

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