`6a^(2) - 13a + 6 =?`

Options

A. `(2a + 3) (3a -2)`
B. `(2z + 1) (3-2x)`
C. `(3a - 2) (2a -3)`
D. `((3a+1) (2a -3)`

Answer

To factor the expression \(6a^2 - 13a + 6\), we will follow these steps:

### Step 1: Identify coefficients
The given quadratic expression is \(6a^2 - 13a + 6\). Here, the coefficients are:
- Coefficient of \(a^2\) (which is \(a^2\)) = 6
- Coefficient of \(a\) (which is \(a\)) = -13
- Constant term = 6

### Step 2: Multiply the coefficient of \(a^2\) with the constant term
We multiply the coefficient of \(a^2\) (which is 6) with the constant term (which is 6):
\[ 6 \times 6 = 36 \]

### Step 3: Find two numbers that multiply to 36 and add to -13
Now, we need to find two numbers that multiply to 36 and add up to -13. The two numbers that satisfy this condition are -9 and -4:
\[ -9 \times -4 = 36 \quad \text{and} \quad -9 + -4 = -13 \]

### Step 4: Rewrite the middle term using the two numbers
We can now rewrite the expression by splitting the middle term (-13a) using -9a and -4a:
\[ 6a^2 - 9a - 4a + 6 \]

### Step 5: Group the terms
Next, we group the terms into two pairs:
\[ (6a^2 - 9a) + (-4a + 6) \]

### Step 6: Factor out the common factors from each group
Now, we factor out the common factors from each group:
- From \(6a^2 - 9a\), we can factor out \(3a\):
\[ 3a(2a - 3) \]
- From \(-4a + 6\), we can factor out \(-2\):
\[ -2(2a - 3) \]

### Step 7: Combine the factored terms
Now, we can combine the factored terms:
\[ 3a(2a - 3) - 2(2a - 3) \]
Since both terms contain \((2a - 3)\), we can factor that out:
\[ (2a - 3)(3a - 2) \]

### Final Answer
Thus, the factorization of the expression \(6a^2 - 13a + 6\) is:
\[ (2a - 3)(3a - 2) \]

---