Solve `6x^(2)-x-2=0`.

Answer

To solve the quadratic equation \(6x^2 - x - 2 = 0\), we will use the method of splitting the middle term. Here’s a step-by-step solution:

### Step 1: Identify the coefficients
The given quadratic equation is in the form \(ax^2 + bx + c = 0\), where:
- \(a = 6\)
- \(b = -1\)
- \(c = -2\)

### Step 2: Calculate the product \(ac\)
We need to find the product \(ac\):
\[ ac = 6 \times (-2) = -12 \]

### Step 3: Find two numbers that multiply to \(ac\) and add to \(b\)
We need to find two numbers that multiply to \(-12\) (the value of \(ac\)) and add to \(-1\) (the value of \(b\)). The numbers that satisfy these conditions are \(3\) and \(-4\) because:
\[ 3 \times (-4) = -12 \quad \text{and} \quad 3 + (-4) = -1 \]

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

### Step 5: Group the terms
Next, we will group the terms:
\[ (6x^2 + 3x) + (-4x - 2) = 0 \]

### Step 6: Factor out the common terms
Now, we factor out the common terms from each group:
\[ 3x(2x + 1) - 2(2x + 1) = 0 \]

### Step 7: Factor by grouping
Now, we can factor out the common binomial factor:
\[ (2x + 1)(3x - 2) = 0 \]

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

### Step 9: Solve for \(x\)
Now, we solve for \(x\) in each equation:
1. From \(2x + 1 = 0\):
\[ 2x = -1 \quad \Rightarrow \quad x = -\frac{1}{2} \]

2. From \(3x - 2 = 0\):
\[ 3x = 2 \quad \Rightarrow \quad x = \frac{2}{3} \]

### Final Solution
The solutions to the equation \(6x^2 - x - 2 = 0\) are:
\[ x = -\frac{1}{2} \quad \text{and} \quad x = \frac{2}{3} \]

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