https://file.aiquickdraw.com/mathbot/file/17266111855103e4ke52g.jpg

I can see the equation from the uploaded image:

The problem is:
\[
\frac{20 - 9x}{4x^2 - 13x - 12} + \frac{x + 2}{4 - x} = 0
\]

The approach taken in the image involves factoring and combining terms. Let me walk through the correct steps to solve this:

### Step 1: Factorize the Denominators
We need to factorize \(4x^2 - 13x - 12\). The factorization yields:
\[
4x^2 - 13x - 12 = (4x + 3)(x - 4)
\]
This will give the first fraction's denominator as \((4x + 3)(x - 4)\).

The second fraction’s denominator is \(4 - x\), which can be rewritten as \(-(x - 4)\) to make the expressions more consistent.

### Step 2: Combine the Fractions
Now, rewrite the equation using these factorizations:
\[
\frac{20 - 9x}{(4x + 3)(x - 4)} + \frac{x + 2}{-(x - 4)} = 0
\]

Next, get a common denominator:
\[
\frac{20 - 9x}{(4x + 3)(x - 4)} - \frac{x + 2}{(x - 4)} = 0
\]

### Step 3: Multiply and Combine Terms
Multiply both sides by the common denominator \((4x + 3)(x - 4)\):
\[
(20 - 9x) - (x + 2)(4x + 3) = 0
\]
Now, expand the terms:
\[
20 - 9x - [(x + 2)(4x + 3)]
\]
Expanding \( (x + 2)(4x + 3) \) gives:
\[
4x^2 + 3x + 8x + 6 = 4x^2 + 11x + 6
\]

Thus, the equation becomes:
\[
20 - 9x - (4x^2 + 11x + 6) = 0
\]

### Step 4: Simplify the Equation
Simplify the terms:
\[
20 - 9x - 4x^2 - 11x - 6 = 0
\]
\[
-4x^2 - 20x + 14 = 0
\]

### Step 5: Solve the Quadratic Equation
To solve the quadratic equation:
\[
4x^2 + 20x - 14 = 0
\]
We can divide the entire equation by 2 to simplify:
\[
2x^2 + 10x - 7 = 0
\]
Now use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \(a = 2\), \(b = 10\), and \(c = -7\).

Substitute the values:
\[
x = \frac{-10 \pm \sqrt{10^2 - 4(2)(-7)}}{2(2)}
\]
\[
x = \frac{-10 \pm \sqrt{100 + 56}}{4}
\]
\[
x = \frac{-10 \pm \sqrt{156}}{4}
\]
\[
x = \frac{-10 \pm 2\sqrt{39}}{4}
\]
\[
x = \frac{-5 \pm \sqrt{39}}{2}
\]

So, the solutions are:
\[
x = \frac{-5 + \sqrt{39}}{2} \quad \text{or} \quad x = \frac{-5 - \sqrt{39}}{2}
\]

Let me know if you'd like more details on any part of the solution or have questions!

---

Here are five related questions to explore further:
1. How do you derive the quadratic formula from a general quadratic equation?
2. What are the conditions for a rational vs. irrational solution in a quadratic equation?
3. How do you factor quadratic expressions efficiently?
4. Can you solve this problem by graphing the equations instead of using algebra?
5. How does multiplying both sides of an equation by the least common denominator work?

**Tip**: Always double-check the sign when rewriting denominators to avoid simple errors in solving!

Solve the equation \(\frac{20 - 9x}{4x^2 - 13x - 12} + \frac{x + 2}{4 - x} = 0\)

Learn how to solve a rational equation involving factoring and the quadratic formula: \(\frac{20 - 9x}{4x^2 - 13x - 12} + \frac{x + 2}{4 - x} = 0\). This problem explores algebraic concepts such as factoring quadratics and solving using the quadratic formula. Suitable for high school students (Grades 9-11).

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