The expression `x^(2)-7x+12 ` is equivalent to :

Options

A. `(x-12 )(x+1)`
B. `(x-4)(x-3)`
C. `(x-4)(x-3)`
D. `(x-6) (x-2)`

Answer

To solve the expression \( x^2 - 7x + 12 \) and find its equivalent factorization, we can follow these steps:

### Step 1: Identify the expression
The given expression is:
\[ x^2 - 7x + 12 \]

### Step 2: Factor the quadratic expression
We need to factor the expression in the form of \( (x - a)(x - b) \) where \( a \) and \( b \) are numbers that satisfy the conditions:
- \( a + b = 7 \) (the coefficient of \( x \) with a negative sign)
- \( ab = 12 \) (the constant term)

### Step 3: Find suitable pairs
We need to find two numbers that multiply to \( 12 \) and add up to \( 7 \). The pairs of factors of \( 12 \) are:
- \( (1, 12) \)
- \( (2, 6) \)
- \( (3, 4) \)

Out of these pairs, \( 3 \) and \( 4 \) add up to \( 7 \):
\[ 3 + 4 = 7 \]

### Step 4: Write the factors
Thus, we can write the expression as:
\[ (x - 3)(x - 4) \]

### Step 5: Verify the factorization
To ensure that our factorization is correct, we can expand \( (x - 3)(x - 4) \):
\[ (x - 3)(x - 4) = x^2 - 4x - 3x + 12 = x^2 - 7x + 12 \]
This confirms that our factorization is correct.

### Conclusion
The expression \( x^2 - 7x + 12 \) is equivalent to:
\[ (x - 3)(x - 4) \]