Factorise :
`x^(2) - 5x + 6`

Answer

To factorise the quadratic expression \( x^2 - 5x + 6 \), we can use the method of splitting the middle term. Here’s a step-by-step solution:

### Step 1: Identify the coefficients
The given expression is \( x^2 - 5x + 6 \). Here, we identify:
- The coefficient of \( x^2 \) (which is \( a \)) = 1
- The coefficient of \( x \) (which is \( b \)) = -5
- The constant term (which is \( c \)) = 6

### Step 2: Multiply \( a \) and \( c \)
Next, we multiply \( a \) and \( c \):
\[ a \cdot c = 1 \cdot 6 = 6 \]

### Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \( 6 \) (the product we found) and add up to \( -5 \) (the coefficient of \( x \)). The numbers that satisfy these conditions are:
- \( -2 \) and \( -3 \) (since \( -2 \times -3 = 6 \) and \( -2 + (-3) = -5 \))

### Step 4: Rewrite the middle term
Now we can rewrite the expression by splitting the middle term using the two numbers we found:
\[ x^2 - 2x - 3x + 6 \]

### Step 5: Group the terms
Next, we group the terms:
\[ (x^2 - 2x) + (-3x + 6) \]

### Step 6: Factor out the common terms in each group
Now we factor out the common factors in each group:
\[ x(x - 2) - 3(x - 2) \]

### Step 7: Factor out the common binomial
Finally, we can factor out the common binomial \( (x - 2) \):
\[ (x - 2)(x - 3) \]

### Conclusion
Thus, the factorised form of the expression \( x^2 - 5x + 6 \) is:
\[ (x - 2)(x - 3) \]

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