Factorise:
`x ^(2) - 20x+ 100`

Answer

To factorise the quadratic expression \( x^2 - 20x + 100 \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the coefficients**:
The expression is in the standard form \( ax^2 + bx + c \) where:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = -20 \) (coefficient of \( x \))
- \( c = 100 \) (constant term)

2. **Find two numbers that multiply to \( ac \) and add to \( b \)**:
We need to find two numbers that multiply to \( a \cdot c = 1 \cdot 100 = 100 \) and add up to \( b = -20 \).
Since we need a sum of \(-20\) and a product of \(100\), we can consider the pairs of factors of \(100\):
- \( (1, 100) \)
- \( (2, 50) \)
- \( (4, 25) \)
- \( (5, 20) \)
- \( (10, 10) \)

The only pair that gives a sum of \(-20\) is \((-10, -10)\).

3. **Rewrite the middle term**:
We can rewrite \(-20x\) as \(-10x - 10x\):
\[ x^2 - 10x - 10x + 100 \]

4. **Group the terms**:
Now, we group the first two terms and the last two terms:
\[ (x^2 - 10x) + (-10x + 100) \]

5. **Factor out the common terms**:
From the first group \( (x^2 - 10x) \), we can factor out \( x \):
\[ x(x - 10) \]
From the second group \( (-10x + 100) \), we can factor out \(-10\):
\[ -10(x - 10) \]

Putting it all together, we have:
\[ x(x - 10) - 10(x - 10) \]

6. **Factor out the common binomial**:
Now we can factor out the common binomial \( (x - 10) \):
\[ (x - 10)(x - 10) \]
or simply:
\[ (x - 10)^2 \]

### Final Answer:
The factorised form of \( x^2 - 20x + 100 \) is:
\[ (x - 10)^2 \]