`(6-x)^(2)-3x `=______

Options

A. `(x-9)(x-4)`
B. `(x-12)(x-3)`
C. `(x+12)(x-3)`
D. `(x-12)(x+3)`

Answer

To factorize the expression \((6-x)^{2} - 3x\), we will follow these steps:

### Step 1: Expand \((6-x)^{2}\)
Using the formula for the square of a binomial, \((a-b)^{2} = a^{2} - 2ab + b^{2}\), we can expand \((6-x)^{2}\):
\[ (6-x)^{2} = 6^{2} - 2 \cdot 6 \cdot x + x^{2} = 36 - 12x + x^{2} \]

### Step 2: Substitute the expansion back into the expression
Now we substitute the expanded form back into the original expression:
\[ (6-x)^{2} - 3x = (36 - 12x + x^{2}) - 3x \]

### Step 3: Combine like terms
Next, we combine the like terms:
\[ 36 - 12x - 3x + x^{2} = x^{2} - 15x + 36 \]

### Step 4: Factor the quadratic expression
Now we need to factor the quadratic expression \(x^{2} - 15x + 36\). We look for two numbers that multiply to \(36\) (the constant term) and add up to \(-15\) (the coefficient of \(x\)). The numbers \(-12\) and \(-3\) satisfy these conditions:
\[ x^{2} - 15x + 36 = (x - 12)(x - 3) \]

### Final Answer
Thus, the factorization of \((6-x)^{2} - 3x\) is:
\[ (6-x)^{2} - 3x = (x - 12)(x - 3) \]

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