Solve the following quadratic equation :
`6a^(2) x^(2) - 7abx - 3b^(2) = 0`

Answer

To solve the quadratic equation \(6a^2 x^2 - 7abx - 3b^2 = 0\), we will use the factorization method. Here are the steps:

### Step 1: Identify the coefficients
The given quadratic equation can be written in the standard form \(Ax^2 + Bx + C = 0\), where:
- \(A = 6a^2\)
- \(B = -7ab\)
- \(C = -3b^2\)

### Step 2: Calculate the product \(AC\)
We need to find \(AC\):
\[ AC = (6a^2)(-3b^2) = -18a^2b^2 \]

### Step 3: Find two numbers that multiply to \(AC\) and add to \(B\)
We need to find two numbers that multiply to \(-18a^2b^2\) and add to \(-7ab\). The numbers are \(-9ab\) and \(2ab\) since:
\[ -9ab \times 2ab = -18a^2b^2 \quad \text{and} \quad -9ab + 2ab = -7ab \]

### Step 4: Rewrite the middle term
Now we rewrite the equation by splitting the middle term using the two numbers found:
\[ 6a^2 x^2 - 9abx + 2abx - 3b^2 = 0 \]

### Step 5: Group the terms
Next, we group the terms:
\[ (6a^2 x^2 - 9abx) + (2abx - 3b^2) = 0 \]

### Step 6: Factor by grouping
Now, we factor each group:
1. From the first group \(6a^2 x^2 - 9abx\), we can factor out \(3ax\):
\[ 3ax(2ax - 3b) \]
2. From the second group \(2abx - 3b^2\), we can factor out \(b\):
\[ b(2ax - 3b) \]

Putting it all together, we have:
\[ 3ax(2ax - 3b) + b(2ax - 3b) = 0 \]

### Step 7: Factor out the common term
Now, we can factor out the common term \((2ax - 3b)\):
\[ (2ax - 3b)(3ax + b) = 0 \]

### Step 8: Set each factor to zero
Now we set each factor to zero:
1. \(2ax - 3b = 0\)
2. \(3ax + b = 0\)

### Step 9: Solve for \(x\)
From the first equation:
\[ 2ax = 3b \implies x = \frac{3b}{2a} \]

From the second equation:
\[ 3ax = -b \implies x = -\frac{b}{3a} \]

### Final Roots
Thus, the roots of the quadratic equation are:
\[ x_1 = \frac{3b}{2a} \quad \text{and} \quad x_2 = -\frac{b}{3a} \]

---