Use the quadratic formula, Solve the following equation for x.
x minus StartFraction 1 Over x plus 2 EndFraction equals negative 2x−
1
x+2=−2


The given equation is:

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

We will solve this using the quadratic formula. Let's first simplify and rearrange the equation.

### Step 1: Multiply both sides by \(x + 2\) to eliminate the denominator.

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

This simplifies to:

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

Distribute on both sides:

\[
x^2 + 2x - 1 = -2x - 4
\]

### Step 2: Move all terms to one side of the equation.

\[
x^2 + 2x - 1 + 2x + 4 = 0
\]

Combine like terms:

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

### Step 3: Apply the quadratic formula.

The quadratic formula is:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \(x^2 + 4x + 3 = 0\), the coefficients are:
- \(a = 1\)
- \(b = 4\)
- \(c = 3\)

Substitute these values into the quadratic formula:

\[
x = \frac{-4 \pm \sqrt{4^2 - 4(1)(3)}}{2(1)}
\]

Simplify:

\[
x = \frac{-4 \pm \sqrt{16 - 12}}{2}
\]

\[
x = \frac{-4 \pm \sqrt{4}}{2}
\]

\[
x = \frac{-4 \pm 2}{2}
\]

### Step 4: Solve for both solutions.

1. \(x_1 = \frac{-4 + 2}{2} = \frac{-2}{2} = -1\)
2. \(x_2 = \frac{-4 - 2}{2} = \frac{-6}{2} = -3\)

### Final Answer:
The solutions are:
\[
x = -1 \quad \text{or} \quad x = -3
\]

Would you like a more detailed explanation of any step?

Here are some related questions to deepen understanding:
1. How does the quadratic formula derive from completing the square?
2. Can you solve \(x^2 + 4x + 5 = 0\) using the quadratic formula?
3. What happens if the discriminant (the part under the square root) is negative?
4. How can factoring be used to solve simpler quadratic equations like \(x^2 + 4x + 3\)?
5. How do you graph \(x^2 + 4x + 3 = 0\) and interpret the solutions?

**Tip**: Always check if you can factor a quadratic equation before applying the quadratic formula—it can save time!

Use the quadratic formula, Solve the following equation for x: x − 1/(x + 2) = -2

Learn how to solve the equation x − 1/(x + 2) = -2 using the quadratic formula. This step-by-step solution covers the core algebraic concepts of quadratic equations and demonstrates how to apply the quadratic formula. Ideal for students in Grades 9-10.

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