Factorise the following :
`x^(2)-9x-36`

Answer

To factorise the expression \( x^2 - 9x - 36 \), we will use the method of middle term splitting. Here’s a step-by-step solution:

### Step 1: Identify the coefficients
The given quadratic expression is in the form \( ax^2 + bx + c \), where:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = -9 \) (coefficient of \( x \))
- \( c = -36 \) (constant term)

### Step 2: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \( ac = 1 \times -36 = -36 \) and add to \( b = -9 \).

The two numbers that satisfy these conditions are \( -12 \) and \( 3 \):
- \( -12 \times 3 = -36 \)
- \( -12 + 3 = -9 \)

### Step 3: Rewrite the middle term
Now, we can rewrite the expression by splitting the middle term using the two numbers found:
\[ x^2 - 12x + 3x - 36 \]

### Step 4: Group the terms
Next, we group the terms:
\[ (x^2 - 12x) + (3x - 36) \]

### Step 5: Factor out the common factors from each group
Now, we factor out the common factors from each group:
- From the first group \( (x^2 - 12x) \), we can factor out \( x \):
\[ x(x - 12) \]
- From the second group \( (3x - 36) \), we can factor out \( 3 \):
\[ 3(x - 12) \]

### Step 6: Combine the factored terms
Now, we can combine the factored terms:
\[ x(x - 12) + 3(x - 12) \]
We can see that \( (x - 12) \) is a common factor:
\[ (x - 12)(x + 3) \]

### Final Answer
Thus, the factorised form of the expression \( x^2 - 9x - 36 \) is:
\[ (x - 12)(x + 3) \]

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