Factorise:
`y ^(2) - 21y + 90`

Answer

To factorise the expression \( y^2 - 21y + 90 \), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic expression is in the form \( ay^2 + by + c \), where:
- \( a = 1 \) (coefficient of \( y^2 \))
- \( b = -21 \) (coefficient of \( y \))
- \( c = 90 \) (constant term)

### Step 2: Multiply \( a \) and \( c \)
We need to multiply \( a \) and \( c \):
\[ a \cdot c = 1 \cdot 90 = 90 \]

### Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)
We are looking for two numbers that multiply to \( 90 \) and add to \( -21 \). The numbers we find are \( -15 \) and \( -6 \):
\[ -15 \times -6 = 90 \quad \text{and} \quad -15 + (-6) = -21 \]

### Step 4: Rewrite the middle term using the two numbers
Now we can rewrite the expression \( y^2 - 21y + 90 \) as:
\[ y^2 - 15y - 6y + 90 \]

### Step 5: Factor by grouping
Next, we group the terms:
\[ (y^2 - 15y) + (-6y + 90) \]

Now, we factor out the common factors in each group:
\[ y(y - 15) - 6(y - 15) \]

### Step 6: Factor out the common binomial
Now we can factor out the common binomial \( (y - 15) \):
\[ (y - 15)(y - 6) \]

### Final Result
Thus, the factorised form of the expression \( y^2 - 21y + 90 \) is:
\[ (y - 15)(y - 6) \]

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