Factorise:
`x ^(2) - 22x + 117`

Answer

To factorise the expression \( x^2 - 22x + 117 \), we will follow these steps:

### Step 1: Identify the coefficients
The given quadratic expression is in the form \( ax^2 + bx + c \), where:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = -22 \) (coefficient of \( x \))
- \( c = 117 \) (constant term)

### Step 2: Multiply \( a \) and \( c \)
We need to multiply the coefficient of \( x^2 \) (which is \( a = 1 \)) with the constant term \( c = 117 \):
\[ 1 \times 117 = 117 \]

### Step 3: Find two numbers that multiply to \( c \) and add to \( b \)
We need to find two numbers that multiply to \( 117 \) and add up to \( -22 \). Since both numbers need to be negative (to add up to a negative number), we can list the pairs of factors of \( 117 \):
- \( -1 \) and \( -117 \)
- \( -3 \) and \( -39 \)
- \( -9 \) and \( -13 \)

Among these pairs, we see that:
\[ -9 + (-13) = -22 \quad \text{and} \quad -9 \times -13 = 117 \]

### Step 4: Rewrite the middle term
Now, we can rewrite the expression \( x^2 - 22x + 117 \) using the two numbers we found:
\[ x^2 - 9x - 13x + 117 \]

### Step 5: Factor by grouping
Next, we will group the terms:
\[ (x^2 - 9x) + (-13x + 117) \]
Now, factor out the common terms in each group:
\[ x(x - 9) - 13(x - 9) \]

### Step 6: Factor out the common binomial
Now we can factor out the common binomial \( (x - 9) \):
\[ (x - 9)(x - 13) \]

### Final Result
Thus, the factorised form of \( x^2 - 22x + 117 \) is:
\[ (x - 9)(x - 13) \]

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