Factorise:
`x ^(2) - 9x + 20`

Answer

To factorise the expression \( x^2 - 9x + 20 \), we can follow these steps:

### Step 1: Identify coefficients
The expression is in the standard quadratic form \( ax^2 + bx + c \), where:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = -9 \) (coefficient of \( x \))
- \( c = 20 \) (constant term)

### Step 2: Multiply \( a \) and \( c \)
We need to multiply \( a \) and \( c \):
\[ a \cdot c = 1 \cdot 20 = 20 \]

### Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \( 20 \) and add up to \( -9 \). The numbers that satisfy this condition are:
- \( -4 \) and \( -5 \) (since \( -4 \times -5 = 20 \) and \( -4 + -5 = -9 \))

### Step 4: Rewrite the middle term
We can rewrite the expression by splitting the middle term using the two numbers found:
\[ x^2 - 4x - 5x + 20 \]

### Step 5: Group the terms
Now, we group the terms in pairs:
\[ (x^2 - 4x) + (-5x + 20) \]

### Step 6: Factor out the common terms
Now, we factor out the common factors from each group:
- From \( x^2 - 4x \), we can factor out \( x \):
\[ x(x - 4) \]
- From \( -5x + 20 \), we can factor out \( -5 \):
\[ -5(x - 4) \]

### Step 7: Combine the factors
Now we can combine the factored terms:
\[ x(x - 4) - 5(x - 4) \]
This can be factored further as:
\[ (x - 4)(x - 5) \]

### Final Answer
Thus, the factorised form of \( x^2 - 9x + 20 \) is:
\[ (x - 4)(x - 5) \]

---