Solve the following quations by using qardratic formula:
`12abx^(2)-(9a^(2)-8b^(2))x-6ab=0`

Answer

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

### Step 1: Identify the coefficients
The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \). Here, we can identify:
- \( a = 12ab \)
- \( b = -(9a^2 - 8b^2) = -9a^2 + 8b^2 \)
- \( c = -6ab \)

### Step 2: Write the quadratic formula
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

### Step 3: Substitute the coefficients into the formula
Substituting the values of \( a \), \( b \), and \( c \) into the quadratic formula:
\[ x = \frac{-(-9a^2 + 8b^2) \pm \sqrt{(-9a^2 + 8b^2)^2 - 4(12ab)(-6ab)}}{2(12ab)} \]
This simplifies to:
\[ x = \frac{9a^2 - 8b^2 \pm \sqrt{(-9a^2 + 8b^2)^2 + 288a^2b^2}}{24ab} \]

### Step 4: Simplify the discriminant
Now, we need to calculate the discriminant:
\[ (-9a^2 + 8b^2)^2 + 288a^2b^2 \]
Calculating \( (-9a^2 + 8b^2)^2 \):
\[ = 81a^4 - 144a^2b^2 + 64b^4 \]
Adding \( 288a^2b^2 \):
\[ 81a^4 - 144a^2b^2 + 64b^4 + 288a^2b^2 = 81a^4 + 144a^2b^2 + 64b^4 \]
This can be factored as:
\[ (9a^2 + 8b^2)^2 \]

### Step 5: Substitute back into the formula
Now substituting back into the formula:
\[ x = \frac{9a^2 - 8b^2 \pm (9a^2 + 8b^2)}{24ab} \]

### Step 6: Solve for the two possible values of \( x \)
1. **First value of \( x \)**:
\[ x_1 = \frac{(9a^2 - 8b^2) + (9a^2 + 8b^2)}{24ab} = \frac{18a^2}{24ab} = \frac{3a}{4b} \]

2. **Second value of \( x \)**:
\[ x_2 = \frac{(9a^2 - 8b^2) - (9a^2 + 8b^2)}{24ab} = \frac{-16b^2}{24ab} = -\frac{2b}{3a} \]

### Final Solution
The solutions to the equation \( 12abx^2 - (9a^2 - 8b^2)x - 6ab = 0 \) are:
\[ x_1 = \frac{3a}{4b} \quad \text{and} \quad x_2 = -\frac{2b}{3a} \]

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