Factorise:
`x^(2) -7x+12`

Answer

To factorise the quadratic expression \( x^2 - 7x + 12 \), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic expression is in the form \( ax^2 + bx + c \). Here, we have:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = -7 \) (coefficient of \( x \))
- \( c = 12 \) (constant term)

### Step 2: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that:
- Multiply to \( ac = 1 \times 12 = 12 \)
- Add to \( b = -7 \)

### Step 3: Determine the numbers
The two numbers that satisfy these conditions are:
- \( -3 \) and \( -4 \)
- Because \( -3 \times -4 = 12 \) (product)
- And \( -3 + (-4) = -7 \) (sum)

### Step 4: Rewrite the middle term
We can now rewrite the expression by splitting the middle term using the numbers we found:
\[ x^2 - 3x - 4x + 12 \]

### Step 5: Factor by grouping
Next, we group the terms:
\[ (x^2 - 3x) + (-4x + 12) \]
Now, factor out the common factors in each group:
\[ x(x - 3) - 4(x - 3) \]

### Step 6: Factor out the common binomial
Now we can factor out the common binomial \( (x - 3) \):
\[ (x - 3)(x - 4) \]

### Final Result
Thus, the factorised form of the expression \( x^2 - 7x + 12 \) is:
\[ (x - 3)(x - 4) \]