Factorise:
`x ^(2) - 10x + 24`

Answer

To factorise the expression \( x^2 - 10x + 24 \), we will follow these steps:

### Step 1: Identify the coefficients
The given quadratic expression is \( x^2 - 10x + 24 \).
- Coefficient of \( x^2 \) (let's call it \( a \)) = 1
- Coefficient of \( x \) (let's call it \( b \)) = -10
- Constant term (let's call it \( c \)) = 24

### Step 2: Multiply \( a \) and \( c \)
We will multiply the coefficient of \( x^2 \) (which is 1) with the constant term (which is 24):
\[ a \cdot c = 1 \cdot 24 = 24 \]

### Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to 24 and add up to -10. The numbers that satisfy these conditions are -6 and -4 because:
\[ -6 \cdot -4 = 24 \quad \text{and} \quad -6 + -4 = -10 \]

### Step 4: Rewrite the middle term
Now we can rewrite the expression \( x^2 - 10x + 24 \) by splitting the middle term using the numbers we found:
\[ x^2 - 6x - 4x + 24 \]

### Step 5: Group the terms
Next, we will group the terms:
\[ (x^2 - 6x) + (-4x + 24) \]

### Step 6: Factor out the common terms
Now we factor out the common factors from each group:
1. From \( x^2 - 6x \), we can factor out \( x \):
\[ x(x - 6) \]
2. From \( -4x + 24 \), we can factor out -4:
\[ -4(x - 6) \]

### Step 7: Combine the factors
Now we can combine the factored terms:
\[ x(x - 6) - 4(x - 6) \]
This can be factored further as:
\[ (x - 6)(x - 4) \]

### Final Answer
Thus, the factorised form of \( x^2 - 10x + 24 \) is:
\[ (x - 6)(x - 4) \]

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