Let's solve the equation:

\[
\frac{45}{x + 6} - \frac{21}{x + 18} = 1
\]

### Step 1: Find a common denominator
The common denominator between \( x + 6 \) and \( x + 18 \) is \( (x + 6)(x + 18) \).

### Step 2: Rewrite the equation with the common denominator
\[
\frac{45(x + 18) - 21(x + 6)}{(x + 6)(x + 18)} = 1
\]

### Step 3: Simplify the numerator
Expand both terms in the numerator:
\[
45(x + 18) = 45x + 810
\]
\[
21(x + 6) = 21x + 126
\]

Now, subtract:
\[
45x + 810 - (21x + 126) = 45x + 810 - 21x - 126 = 24x + 684
\]

### Step 4: Rewrite the equation
Now, the equation is:
\[
\frac{24x + 684}{(x + 6)(x + 18)} = 1
\]

### Step 5: Cross-multiply to eliminate the fraction
\[
24x + 684 = (x + 6)(x + 18)
\]

### Step 6: Expand and simplify the quadratic equation
First, expand \( (x + 6)(x + 18) \):
\[
(x + 6)(x + 18) = x^2 + 18x + 6x + 108 = x^2 + 24x + 108
\]

So, the equation becomes:
\[
24x + 684 = x^2 + 24x + 108
\]

### Step 7: Move all terms to one side of the equation
\[
0 = x^2 + 24x + 108 - 24x - 684
\]

Simplify:
\[
x^2 - 576 = 0
\]

### Step 8: Solve for \( x \)
\[
x^2 = 576
\]
\[
x = \pm \sqrt{576}
\]
\[
x = \pm 24
\]

### Step 9: Check for extraneous solutions
We must ensure that these solutions do not make the original denominators zero.

- If \( x = 24 \), the denominators \( x + 6 \) and \( x + 18 \) are 30 and 42, respectively (both non-zero).
- If \( x = -24 \), the denominators \( x + 6 \) and \( x + 18 \) are -18 and -6, respectively (both non-zero).

Thus, both solutions \( x = 24 \) and \( x = -24 \) are valid.

### Final Answer:
\[
x = 24 \quad \text{or} \quad x = -24
\]

Would you like further details or have any questions?

Here are 5 related questions you might find interesting:
1. How do you solve equations with variables in the denominator?
2. What happens if the solution makes the denominator zero?
3. Can you explain how to factor quadratic equations?
4. What is the significance of cross-multiplication in solving rational equations?
5. How do you check for extraneous solutions in rational equations?

**Tip:** Always check your solutions by substituting them back into the original equation, especially when dealing with fractions or square roots.

Learn how to solve a rational equation involving quadratics: 45/(x + 6) - 21/(x + 18) = 1. This problem involves finding a common denominator, simplifying expressions, and solving a quadratic equation. Suitable for students in Grades 9-11.

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