https://file.aiquickdraw.com/mathbot/file/1725776055689z9u2c5no.jpg

The equation provided in the image is:

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

To solve this equation, follow these steps:

### Step 1: Eliminate the denominator
Multiply both sides of the equation by \(x - 5\) (assuming \(x \neq 5\) to avoid division by zero):

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

### Step 2: Simplify the equation
Distribute on the right side:

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

### Step 3: Move all terms to one side
Subtract \(x\) and add 5 to both sides:

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

This simplifies to:

\[
x^2 - 25 = 0
\]

### Step 4: Factor the quadratic equation
Factor \(x^2 - 25\) as a difference of squares:

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

### Step 5: Solve for \(x\)
Set each factor equal to zero:

1. \(x - 5 = 0 \)  gives \( x = 5 \)
2. \(x + 5 = 0 \) gives \( x = -5 \)

### Step 6: Check for extraneous solutions
Substitute \(x = 5\) into the original equation:

\[
\frac{5^2 + 5 - 30}{5 - 5} = \frac{0}{0}
\]

Since this leads to an undefined expression (division by zero), \(x = 5\) is not a valid solution.

Therefore, the only solution is:

\[
\boxed{x = -5}
\]

Would you like any additional details or have any questions about this solution? 

Here are 5 related questions to consider:

1. How can you identify when to use factoring to solve a quadratic equation?
2. What is the difference between solving a quadratic equation by factoring and by using the quadratic formula?
3. How do you handle equations that result in division by zero?
4. Can you have more than one valid solution when solving a quadratic equation?
5. What other methods can be used to solve quadratic equations?

**Tip:** Always check for extraneous solutions when working with rational equations, especially when the equation involves variable expressions in the denominator.

Solve the equation (x^2 + x - 30) / (x - 5) = 1

This guide explains how to solve the rational equation (x^2 + x - 30) / (x - 5) = 1, utilizing algebraic methods including factoring and the difference of squares theorem. Suitable for high school students studying advanced algebra.

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