Factorise : (i) `x^2 - 9x + 20`

Answer

To factorise the quadratic expression \(x^2 - 9x + 20\), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic expression is in the form \(ax^2 + bx + c\). Here, \(a = 1\), \(b = -9\), and \(c = 20\).

### Step 2: Find two numbers that multiply to \(c\) and add to \(b\)
We need to find two numbers that multiply to \(c\) (which is 20) and add up to \(b\) (which is -9).

The factors of 20 are:
- \(1 \times 20\)
- \(2 \times 10\)
- \(4 \times 5\)

Now, we check their sums:
- \(1 + 20 = 21\) (not -9)
- \(2 + 10 = 12\) (not -9)
- \(4 + 5 = 9\) (not -9)

Next, we consider negative factors:
- \(-4\) and \(-5\) multiply to \(20\) and add to \(-9\).

### Step 3: Rewrite the expression using the factors
We can rewrite the middle term \(-9x\) using the factors \(-4\) and \(-5\):
\[ x^2 - 4x - 5x + 20 \]

### Step 4: Group the terms
Now, we group the terms:
\[ (x^2 - 4x) + (-5x + 20) \]

### Step 5: Factor by grouping
Now, we factor out the common factors from each group:
\[ x(x - 4) - 5(x - 4) \]

### Step 6: Factor out the common binomial
Now, we can factor out the common binomial \((x - 4)\):
\[ (x - 4)(x - 5) \]

### Final Answer
Thus, the factorised form of the expression \(x^2 - 9x + 20\) is:
\[ (x - 4)(x - 5) \]

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