Factorise:
`x ^(2) - 5x - 24`

Options

A. `(x-8)(x-3)`
B. `(x-8)(x+3)`
C. `(x+8)(x+3)`
D. none of these

Answer

To factorise the expression \( x^2 - 5x - 24 \), we can follow these steps:

### Step 1: Identify the coefficients
We start with the quadratic expression:
\[ x^2 - 5x - 24 \]
Here, the coefficient of \( x^2 \) is \( 1 \), the coefficient of \( x \) is \( -5 \), and the constant term is \( -24 \).

### Step 2: Multiply the coefficient of \( x^2 \) by the constant term
Next, we multiply the coefficient of \( x^2 \) (which is \( 1 \)) by the constant term (which is \( -24 \)):
\[ 1 \times (-24) = -24 \]

### Step 3: Find two numbers that multiply to \(-24\) and add to \(-5\)
Now, we need to find two numbers that multiply to \(-24\) and add to \(-5\). The pairs of factors of \(-24\) are:
- \( 1 \) and \(-24\)
- \(-1\) and \(24\)
- \(2\) and \(-12\)
- \(-2\) and \(12\)
- \(3\) and \(-8\)
- \(-3\) and \(8\)

Among these pairs, the numbers \(3\) and \(-8\) satisfy our conditions:
\[ 3 + (-8) = -5 \quad \text{and} \quad 3 \times (-8) = -24 \]

### Step 4: Rewrite the middle term using the two numbers
We can now rewrite the expression by splitting the middle term \(-5x\) into \(3x\) and \(-8x\):
\[ x^2 + 3x - 8x - 24 \]

### Step 5: Group the terms
Next, we group the terms:
\[ (x^2 + 3x) + (-8x - 24) \]

### Step 6: Factor out the common terms from each group
Now, we factor out the common factors from each group:
\[ x(x + 3) - 8(x + 3) \]

### Step 7: Factor out the common binomial
We notice that \( (x + 3) \) is a common factor:
\[ (x + 3)(x - 8) \]

### Final Result
Thus, the factorised form of the expression \( x^2 - 5x - 24 \) is:
\[ (x + 3)(x - 8) \]

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