https://file.aiquickdraw.com/mathbot/file/1726119427867hp0i1eqo.jpg

The given equation from the image is:

\[
\frac{4}{x-3} - \frac{1}{6 + x - x^2} = 0
\]

### Steps to solve this equation:

1. **Combine the fractions**: We want to bring both fractions over a common denominator. The denominators are \(x - 3\) and \(6 + x - x^2\).

\[
\frac{4}{x-3} = \frac{1}{6 + x - x^2}
\]

2. **Cross-multiply**: Since the fractions are equal, cross-multiplying gives us:

\[
4(6 + x - x^2) = 1(x - 3)
\]

3. **Expand both sides**:
   - On the left-hand side: \( 4(6 + x - x^2) = 24 + 4x - 4x^2 \)
   - On the right-hand side: \( 1(x - 3) = x - 3 \)

So, the equation becomes:

\[
24 + 4x - 4x^2 = x - 3
\]

4. **Move all terms to one side**:

\[
24 + 4x - 4x^2 - x + 3 = 0
\]

Simplifying:

\[
27 + 3x - 4x^2 = 0
\]

5. **Rearrange into standard quadratic form**:

\[
-4x^2 + 3x + 27 = 0
\]

Multiply through by \(-1\) to simplify:

\[
4x^2 - 3x - 27 = 0
\]

6. **Solve the quadratic equation** using the quadratic formula:

\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(4)(-27)}}{2(4)}
\]

\[
x = \frac{3 \pm \sqrt{9 + 432}}{8}
\]

\[
x = \frac{3 \pm \sqrt{441}}{8}
\]

\[
x = \frac{3 \pm 21}{8}
\]

This gives two solutions:

\[
x = \frac{3 + 21}{8} = \frac{24}{8} = 3
\]
and
\[
x = \frac{3 - 21}{8} = \frac{-18}{8} = -\frac{9}{4}
\]

### Exclude extraneous solutions:
Since \(x = 3\) would make the denominator \(x - 3 = 0\), we must exclude it.

Thus, the solution is:

\[
x = -\frac{9}{4}
\]

Would you like further details on any step?

### 5 Related Questions:
1. How do you determine extraneous solutions in equations with fractions?
2. What are the steps to cross-multiply and solve rational equations?
3. How does the quadratic formula work, and when should it be applied?
4. What strategies can be used to factor complex quadratics?
5. Why does a solution become extraneous when it makes the denominator zero?

### Tip:
Always check for extraneous solutions in rational equations by substituting back into the original equation.

Solve the equation 4/(x - 3) - 1/(6 + x - x^2) = 0.

Learn how to solve the rational equation 4/(x - 3) - 1/(6 + x - x^2) = 0 using cross-multiplication and the quadratic formula. This solution is suitable for high school students and involves key algebraic techniques.

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