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

Answer

To solve the quadratic equation \(6x^2 - x - 2 = 0\), we can use the method of splitting the middle term. Here are the steps:

### 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 two numbers that multiply to \(ac\) (which is \(6 \times -2 = -12\)) and add to \(b\) (which is \(-1\)).

### Step 3: Find the two numbers
We need to find two numbers that multiply to \(-12\) and add to \(-1\). The numbers are \(-4\) and \(3\) because:
- \(-4 \times 3 = -12\)
- \(-4 + 3 = -1\)

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

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

### Step 6: Factor by grouping
Now we factor out the common factors from each group:
\[ 2x(3x - 2) + 1(3x - 2) = 0 \]
This can be factored as:
\[ (2x + 1)(3x - 2) = 0 \]

### Step 7: Set each factor to zero
Now we set each factor equal to zero:
1. \(2x + 1 = 0\)
2. \(3x - 2 = 0\)

### Step 8: Solve for \(x\)
1. For \(2x + 1 = 0\):
\[ 2x = -1 \implies x = -\frac{1}{2} \]

2. For \(3x - 2 = 0\):
\[ 3x = 2 \implies x = \frac{2}{3} \]

### Step 9: Write the 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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