https://file.aiquickdraw.com/mathbot/file/17246813489053wbuawaf.jpg

The image contains a mathematical equation that needs to be solved. The equation is:

\[
\frac{7}{x-1} - \frac{x-1}{x+3} = 1
\]

### Let's solve this equation step-by-step:

1. **Combine the fractions on the left side:**  
   To combine the fractions, we need to have a common denominator. The common denominator for the fractions is \((x-1)(x+3)\).

\[
\frac{7(x+3)}{(x-1)(x+3)} - \frac{(x-1)^2}{(x-1)(x+3)} = 1
\]

2. **Simplify the numerators:**

   - Distribute the 7 in the first fraction:

\[
\frac{7x + 21}{(x-1)(x+3)} - \frac{(x-1)^2}{(x-1)(x+3)} = 1
\]

   - Expand \((x-1)^2\):

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

So, the equation becomes:

\[
\frac{7x + 21 - (x^2 - 2x + 1)}{(x-1)(x+3)} = 1
\]

3. **Simplify further:**

Combine like terms in the numerator:

\[
\frac{7x + 21 - x^2 + 2x - 1}{(x-1)(x+3)} = 1
\]

This simplifies to:

\[
\frac{-x^2 + 9x + 20}{(x-1)(x+3)} = 1
\]

4. **Clear the fraction by multiplying both sides by \((x-1)(x+3)\):**

\[
-x^2 + 9x + 20 = (x-1)(x+3)
\]

5. **Expand the right-hand side:**

Expand \((x-1)(x+3)\):

\[
(x-1)(x+3) = x^2 + 3x - x - 3 = x^2 + 2x - 3
\]

Now, we have:

\[
-x^2 + 9x + 20 = x^2 + 2x - 3
\]

6. **Move all terms to one side of the equation:**

\[
-x^2 + 9x + 20 - x^2 - 2x + 3 = 0
\]

Simplify:

\[
-2x^2 + 7x + 23 = 0
\]

7. **Factorize the quadratic equation:**

To factorize \(-2x^2 + 7x + 23 = 0\), we can look for factors or use the quadratic formula. In this case, let's rewrite it for easier factorization, or solve using the quadratic formula.

### Quadratic Formula:

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

Here, \(a = -2\), \(b = 7\), and \(c = 23\). Let's plug in the values:

\[
x = \frac{-7 \pm \sqrt{7^2 - 4(-2)(23)}}{2(-2)}
\]

\[
x = \frac{-7 \pm \sqrt{49 + 184}}{-4}
\]

\[
x = \frac{-7 \pm \sqrt{233}}{-4}
\]

The solutions are:

\[
x = \frac{-7 + \sqrt{233}}{-4}, \quad x = \frac{-7 - \sqrt{233}}{-4}
\]

### Final Solutions:
\[
x = \frac{7 - \sqrt{233}}{4}, \quad x = \frac{7 + \sqrt{233}}{4}
\]

These are the roots of the quadratic equation. Factorization isn't straightforward in this case because of the complex square root, so solving using the quadratic formula is the optimal approach.

Would you like more details on any of these steps? 

Here are some related questions:

1. What is the importance of the quadratic formula in solving equations?
2. How do you determine the common denominator when adding fractions with different denominators?
3. What are other methods to solve quadratic equations apart from the quadratic formula?
4. Can the discriminant be used to determine the nature of the roots of the quadratic equation?
5. What are some examples of quadratic equations that can be easily factorized?

**Tip:** When dealing with complex fractions, always look for a common denominator to simplify the equation first.

Learn how to solve the fractional equation (7/(x-1)) - ((x-1)/(x+3)) = 1 step-by-step with detailed explanations. This problem covers fractional equations, quadratic equations, and the quadratic formula. Suitable for advanced high school students.

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