Factorise `x^2-22x+120`

Answer

To factorise the quadratic expression \( x^2 - 22x + 120 \), we will follow these steps:

### Step 1: Identify coefficients
The given quadratic expression is in the form \( ax^2 + bx + c \), where:
- \( a = 1 \)
- \( b = -22 \)
- \( c = 120 \)

### Step 2: Find two numbers that add up to \( b \) and multiply to \( ac \)
We need to find two numbers \( b_1 \) and \( b_2 \) such that:
- \( b_1 + b_2 = b = -22 \)
- \( b_1 \times b_2 = ac = 1 \times 120 = 120 \)

### Step 3: List pairs of factors of 120
The pairs of factors of 120 are:
- \( 1 \times 120 \)
- \( 2 \times 60 \)
- \( 3 \times 40 \)
- \( 4 \times 30 \)
- \( 5 \times 24 \)
- \( 6 \times 20 \)
- \( 8 \times 15 \)
- \( 10 \times 12 \)

### Step 4: Check which pair sums to -22
We need to consider negative pairs since \( b \) is negative:
- \( -10 \) and \( -12 \) are the pair that adds up to \( -22 \) and multiplies to \( 120 \).

### Step 5: Rewrite the middle term
Now, we can rewrite the expression \( x^2 - 22x + 120 \) as:
\[ x^2 - 10x - 12x + 120 \]

### Step 6: Factor by grouping
Next, we group the terms:
\[ (x^2 - 10x) + (-12x + 120) \]
Factoring out the common factors from each group:
\[ x(x - 10) - 12(x - 10) \]

### Step 7: Factor out the common binomial
Now we can factor out the common binomial \( (x - 10) \):
\[ (x - 10)(x - 12) \]

### Final Answer
Thus, the factorization of \( x^2 - 22x + 120 \) is:
\[ (x - 10)(x - 12) \]

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