Find the roots of the quadratic equation `6x^2-x-2=0`.

Options

A. -1/2, 3/2
B. -1/2, 2/3
C. -1/2, -2/3
D. None of these

Explanation

To find the roots of the quadratic equation \(6x^2 - x - 2 = 0\), we can use the method of factorization. Here’s a step-by-step solution:

### Step 1: Write down the quadratic equation
The given quadratic equation is:
\[ 6x^2 - x - 2 = 0 \]

### Step 2: Identify coefficients
In the equation \(ax^2 + bx + c = 0\), we have:
- \(a = 6\)
- \(b = -1\)
- \(c = -2\)

### Step 3: Calculate the product \(ac\)
We need to find two numbers that multiply to \(ac\) (which is \(6 \times -2 = -12\)) and add up to \(b\) (which is \(-1\)).

### Step 4: Find the two numbers
We need two numbers that:
- Multiply to \(-12\)
- Add to \(-1\)

The numbers that satisfy these conditions are \(3\) and \(-4\) because:
\[ 3 \times -4 = -12 \quad \text{and} \quad 3 + (-4) = -1 \]

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

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

### Step 8: Factor out the common binomial
Now we can factor out the common binomial \((2x + 1)\):
\[ (3x - 2)(2x + 1) = 0 \]

### Step 9: Set each factor to zero
Now we set each factor equal to zero:
1. \(3x - 2 = 0\)
2. \(2x + 1 = 0\)

### Step 10: Solve for \(x\)
1. From \(3x - 2 = 0\):
\[ 3x = 2 \implies x = \frac{2}{3} \]

2. From \(2x + 1 = 0\):
\[ 2x = -1 \implies x = -\frac{1}{2} \]

### Step 11: State the roots
The roots of the quadratic equation \(6x^2 - x - 2 = 0\) are:
\[ x = \frac{2}{3} \quad \text{and} \quad x = -\frac{1}{2} \]

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