5. Solve `12abx^2-9a^2x +8b^2x-6ab =0`

Answer

To solve the quadratic equation \( 12abx^2 - 9a^2x + 8b^2x - 6ab = 0 \), we will follow these steps:

### Step 1: Rearranging the Equation
We start with the equation:
\[ 12abx^2 - 9a^2x + 8b^2x - 6ab = 0 \]

### Step 2: Grouping Terms
We can group the terms in pairs:
\[ (12abx^2 - 9a^2x) + (8b^2x - 6ab) = 0 \]

### Step 3: Factoring Out Common Terms
Now, we factor out the common terms from each group:
1. From the first group \( 12abx^2 - 9a^2x \), we can factor out \( 3ax \):
\[ 3ax(4bx - 3a) \]
2. From the second group \( 8b^2x - 6ab \), we can factor out \( 2b \):
\[ 2b(4bx - 3a) \]

So, we can rewrite the equation as:
\[ 3ax(4bx - 3a) + 2b(4bx - 3a) = 0 \]

### Step 4: Factoring the Entire Expression
Now, we can factor out the common binomial \( (4bx - 3a) \):
\[ (4bx - 3a)(3ax + 2b) = 0 \]

### Step 5: Setting Each Factor to Zero
Now, we set each factor to zero:
1. \( 4bx - 3a = 0 \)
2. \( 3ax + 2b = 0 \)

### Step 6: Solving for \( x \)
1. From \( 4bx - 3a = 0 \):
\[ 4bx = 3a \implies x = \frac{3a}{4b} \]

2. From \( 3ax + 2b = 0 \):
\[ 3ax = -2b \implies x = -\frac{2b}{3a} \]

### Final Solutions
Thus, the solutions for \( x \) are:
\[ x = \frac{3a}{4b} \quad \text{and} \quad x = -\frac{2b}{3a} \]

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