https://file.aiquickdraw.com/mathbot/file/1725413975344e92l4gca.jpg

To solve the equation given in the image:

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

We can follow these steps:

1. **Multiply both sides by the denominator** \(x - 5\) to eliminate the fraction:

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

This simplifies to:

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

2. **Move all terms to one side of the equation** by subtracting \(x\) and adding 5 from both sides:

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

This simplifies to:

\[
x^2 - 25 = 0
\]

3. **Factor the quadratic equation**:

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

4. **Solve for \(x\)** by setting each factor equal to zero:

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

So:

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

5. **Check for extraneous solutions** by substituting these values back into the original equation:

- For \(x = 5\), the denominator becomes zero, which is undefined. So, \(x = 5\) is not a valid solution.
- For \(x = -5\), substitute into the original equation:

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

This shows that \(x = -5\) is indeed a valid solution.

**Final Answer**: The solution to the equation is \(x = -5\).

Would you like further details or have any questions?

Here are 5 related questions:

1. How do you solve quadratic equations by factoring?
2. What are extraneous solutions, and why do they occur?
3. How would you solve the equation if it were a different type of equation, like a cubic?
4. What other methods can you use to solve quadratic equations?
5. How can you verify your solution graphically?

**Tip**: Always check for extraneous solutions when solving equations that involve fractions or square roots, as the solutions may not always satisfy the original equation.

Solve the equation \(\frac{x^2 + x - 30}{x - 5} = 1\).

Learn how to solve the rational equation \(\frac{x^2 + x - 30}{x - 5} = 1\) with a detailed step-by-step solution. This problem covers key algebraic concepts, including factoring, rational expressions, and checking for extraneous solutions. Suitable for students in Grades 9-11.

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