(x+5)/(x-2) = [(5)/(x+2)] + [(28)/(x^2 - 4)] 

We are given the equation:

\[
\frac{x+5}{x-2} = \frac{5}{x+2} + \frac{28}{x^2 - 4}
\]

### Step 1: Simplify the right-hand side
Notice that \(x^2 - 4\) is a difference of squares, which can be factored as:

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

So the equation becomes:

\[
\frac{x+5}{x-2} = \frac{5}{x+2} + \frac{28}{(x-2)(x+2)}
\]

### Step 2: Find a common denominator on the right-hand side
The common denominator for the terms on the right-hand side is \((x-2)(x+2)\). We will rewrite the fractions on the right-hand side with this common denominator:

\[
\frac{5}{x+2} = \frac{5(x-2)}{(x-2)(x+2)}
\]

Now the equation is:

\[
\frac{x+5}{x-2} = \frac{5(x-2) + 28}{(x-2)(x+2)}
\]

### Step 3: Simplify the right-hand side
Distribute the 5 in the numerator:

\[
5(x-2) = 5x - 10
\]

So the right-hand side becomes:

\[
\frac{5x - 10 + 28}{(x-2)(x+2)} = \frac{5x + 18}{(x-2)(x+2)}
\]

### Step 4: Cross-multiply
Now that both sides of the equation have a denominator, we can cross-multiply:

\[
(x+5)(x+2) = (x-2)(5x+18)
\]

### Step 5: Expand both sides
Expand both sides:

**Left-hand side:**

\[
(x+5)(x+2) = x^2 + 2x + 5x + 10 = x^2 + 7x + 10
\]

**Right-hand side:**

\[
(x-2)(5x+18) = x(5x+18) - 2(5x+18) = 5x^2 + 18x - 10x - 36 = 5x^2 + 8x - 36
\]

### Step 6: Set the equation to 0
Now, set the equation to 0:

\[
x^2 + 7x + 10 = 5x^2 + 8x - 36
\]

Move all terms to one side:

\[
0 = 5x^2 + 8x - 36 - x^2 - 7x - 10
\]

Simplify:

\[
0 = 4x^2 + x - 46
\]

### Step 7: Solve the quadratic equation
Now, we solve the quadratic equation:

\[
4x^2 + x - 46 = 0
\]

We can use the quadratic formula:

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

Here, \(a = 4\), \(b = 1\), and \(c = -46\).

\[
x = \frac{-1 \pm \sqrt{1^2 - 4(4)(-46)}}{2(4)} = \frac{-1 \pm \sqrt{1 + 736}}{8} = \frac{-1 \pm \sqrt{737}}{8}
\]

Thus, the solutions are:

\[
x = \frac{-1 + \sqrt{737}}{8} \quad \text{or} \quad x = \frac{-1 - \sqrt{737}}{8}
\]

These are the two solutions for \(x\).

Would you like to discuss any part of the solution or the quadratic formula?

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Here are five related questions to explore further:
1. How do you factor the difference of squares?
2. What are the steps to simplify expressions with complex denominators?
3. How can we solve quadratic equations without the quadratic formula?
4. What are the applications of rational equations in real-world problems?
5. How does cross-multiplication work when solving rational equations?

**Tip:** Always look for factoring opportunities when working with quadratic expressions. It simplifies the problem and makes solving easier.

(x+5)/(x-2) = [(5)/(x+2)] + [(28)/(x^2 - 4)]

Discover how to solve the rational equation (x+5)/(x-2) = (5)/(x+2) + (28)/(x^2-4) using factoring and the quadratic formula. This detailed step-by-step solution explains algebraic concepts like cross-multiplication, difference of squares, and solving quadratic equations. Ideal for students in Grades 9-11.

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