Find the roots of equation `6x^2-13x+6 = 0`

Options

A. `(18)/(14), (8)/(12)`
B. `(18)/(12), (8)/(12)`
C. `(18)/(12), (4)/(12)`
D. `(8)/(12), (8)/(12)`

Explanation

To find the roots of the quadratic equation \(6x^2 - 13x + 6 = 0\), we will use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

### Step 1: Identify the coefficients
In the equation \(6x^2 - 13x + 6 = 0\), we can identify the coefficients:
- \(a = 6\)
- \(b = -13\)
- \(c = 6\)

### Step 2: Calculate the discriminant
The discriminant \(D\) is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \(a\), \(b\), and \(c\):
\[ D = (-13)^2 - 4 \cdot 6 \cdot 6 \]
Calculating \(D\):
\[ D = 169 - 144 = 25 \]

### Step 3: Apply the quadratic formula
Now that we have the discriminant, we can substitute \(a\), \(b\), and \(D\) into the quadratic formula:
\[ x = \frac{-(-13) \pm \sqrt{25}}{2 \cdot 6} \]
This simplifies to:
\[ x = \frac{13 \pm 5}{12} \]

### Step 4: Calculate the two possible values for \(x\)
Now we will calculate the two roots:
1. For the positive case:
\[ x_1 = \frac{13 + 5}{12} = \frac{18}{12} = \frac{3}{2} \]
2. For the negative case:
\[ x_2 = \frac{13 - 5}{12} = \frac{8}{12} = \frac{2}{3} \]

### Final Answer
The roots of the equation \(6x^2 - 13x + 6 = 0\) are:
\[ x_1 = \frac{3}{2}, \quad x_2 = \frac{2}{3} \]