Factorise:
`q ^(2) - 10q + 21`

Answer

To factorise the expression \( q^2 - 10q + 21 \), we will follow these steps:

### Step 1: Identify the coefficients
The expression is in the standard quadratic form \( ax^2 + bx + c \), where:
- \( a = 1 \) (coefficient of \( q^2 \))
- \( b = -10 \) (coefficient of \( q \))
- \( c = 21 \) (constant term)

**Hint:** Identify the coefficients of the quadratic expression to understand what you are working with.

### Step 2: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \( a \cdot c = 1 \cdot 21 = 21 \) and add to \( b = -10 \).

The two numbers that meet these criteria are \( -3 \) and \( -7 \):
- \( -3 \times -7 = 21 \)
- \( -3 + -7 = -10 \)

**Hint:** Look for pairs of factors of the constant term that also add up to the coefficient of \( q \).

### Step 3: Rewrite the middle term
Using the two numbers found, we can rewrite the expression:
\[ q^2 - 10q + 21 = q^2 - 3q - 7q + 21 \]

**Hint:** Break down the middle term into two terms using the numbers you found.

### Step 4: Group the terms
Now we group the terms:
\[ (q^2 - 3q) + (-7q + 21) \]

**Hint:** Group the terms in pairs to prepare for factoring.

### Step 5: Factor out the common terms
Now, we factor out the common factors from each group:
\[ q(q - 3) - 7(q - 3) \]

**Hint:** Look for common factors in each group and factor them out.

### Step 6: Factor by grouping
Now we can factor out the common binomial factor \( (q - 3) \):
\[ (q - 3)(q - 7) \]

**Hint:** Once you have a common factor, factor it out to simplify the expression.

### Final Answer
Thus, the factorised form of \( q^2 - 10q + 21 \) is:
\[ (q - 3)(q - 7) \]