https://file.aiquickdraw.com/mathbot/file/1724508890515b8dz3c8b.jpg

https://file.aiquickdraw.com/mathbot/file/1724508890516qhj5zylr.jpg

Let's solve the given equation:

\[
\frac{x^2 + x - 30}{x - 5} = 1
\]

### Step 1: Eliminate the denominator
To eliminate the fraction, multiply both sides of the equation by \(x - 5\):

\[
x^2 + x - 30 = 1(x - 5)
\]

### Step 2: Expand and simplify
Distribute the \(1\) on the right-hand side:

\[
x^2 + x - 30 = x - 5
\]

Now, subtract \(x\) and add \(5\) from both sides to bring all terms to the left-hand side:

\[
x^2 + x - 30 - x + 5 = 0
\]

Simplify:

\[
x^2 - 25 = 0
\]

### Step 3: Solve the quadratic equation
This equation can be factored:

\[
(x - 5)(x + 5) = 0
\]

So, the solutions are:

\[
x - 5 = 0 \quad \text{or} \quad x + 5 = 0
\]

\[
x = 5 \quad \text{or} \quad x = -5
\]

### Step 4: Check for extraneous solutions
However, we must check if \(x = 5\) leads to a valid solution since it could make the denominator zero:

1. For \(x = 5\), the original equation becomes undefined because the denominator \(x - 5\) would be zero.
2. For \(x = -5\), substitute back into the original equation:

\[
\frac{(-5)^2 - 5 - 30}{-5 - 5} = \frac{25 - 5 - 30}{-10} = \frac{-10}{-10} = 1
\]

So, \(x = -5\) is the only valid solution.

### Final Answer:
\[
x = -5
\]

Would you like more details, or do you have any questions?

---

**Related Questions:**

1. How do you identify extraneous solutions in rational equations?
2. What happens when you multiply or divide by an expression that could be zero?
3. How can you factor a quadratic equation like \(x^2 - 25 = 0\) efficiently?
4. Why is it important to check solutions in the original equation?
5. How does the graph of a rational function relate to its equation?

**Tip:** Always check your solutions in the original equation, especially when dealing with rational equations, to avoid extraneous solutions.

Learn how to solve the rational equation (x^2 + x - 30) / (x - 5) = 1 with detailed step-by-step solutions and explanations. This problem involves quadratic equations, rational expressions, and the importance of checking for extraneous solutions. Suitable for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation