`A={x|3x^2-7x-6=0},B={x|6x^2-5x-6=0}`. Find `A nn B`

Options

A. `{-2/3}`
B. `{-1/3}`
C. `{-4/3}`
D. `{-5/3}`

Answer

To solve the problem, we need to find the intersection of two sets A and B, where:

- Set A is defined by the roots of the equation \(3x^2 - 7x - 6 = 0\).
- Set B is defined by the roots of the equation \(6x^2 - 5x - 6 = 0\).

### Step 1: Find the roots of the equation for Set A
We will solve the quadratic equation \(3x^2 - 7x - 6 = 0\) using the middle-term splitting method.

1. **Identify the coefficients**: Here, \(a = 3\), \(b = -7\), and \(c = -6\).
2. **Calculate the product \(ac\)**: \(ac = 3 \times -6 = -18\).
3. **Find two numbers that multiply to \(-18\) and add to \(-7\)**: The numbers are \(-9\) and \(2\).
4. **Rewrite the equation**:
\[ 3x^2 - 9x + 2x - 6 = 0 \]
5. **Factor by grouping**:
\[ 3x(x - 3) + 2(x - 3) = 0 \]
\[ (3x + 2)(x - 3) = 0 \]
6. **Set each factor to zero**:
\[ 3x + 2 = 0 \quad \Rightarrow \quad x = -\frac{2}{3} \]
\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]
7. **Thus, Set A**:
\[ A = \left\{-\frac{2}{3}, 3\right\} \]

### Step 2: Find the roots of the equation for Set B
Now, we will solve the quadratic equation \(6x^2 - 5x - 6 = 0\) using the same method.

1. **Identify the coefficients**: Here, \(a = 6\), \(b = -5\), and \(c = -6\).
2. **Calculate the product \(ac\)**: \(ac = 6 \times -6 = -36\).
3. **Find two numbers that multiply to \(-36\) and add to \(-5\)**: The numbers are \(-9\) and \(4\).
4. **Rewrite the equation**:
\[ 6x^2 - 9x + 4x - 6 = 0 \]
5. **Factor by grouping**:
\[ 3x(2x - 3) + 2(2x - 3) = 0 \]
\[ (2x - 3)(3x + 2) = 0 \]
6. **Set each factor to zero**:
\[ 2x - 3 = 0 \quad \Rightarrow \quad x = \frac{3}{2} \]
\[ 3x + 2 = 0 \quad \Rightarrow \quad x = -\frac{2}{3} \]
7. **Thus, Set B**:
\[ B = \left\{-\frac{2}{3}, \frac{3}{2}\right\} \]

### Step 3: Find the intersection of sets A and B
Now we need to find the intersection \(A \cap B\).

- Set A: \(\left\{-\frac{2}{3}, 3\right\}\)
- Set B: \(\left\{-\frac{2}{3}, \frac{3}{2}\right\}\)

The common element in both sets is \(-\frac{2}{3}\).

### Final Answer
\[ A \cap B = \left\{-\frac{2}{3}\right\} \]