Factorise : (ii) `y^2 + 5y - 24`

Answer

To factorise the expression \( y^2 + 5y - 24 \), we will follow these steps:

### Step 1: Identify the coefficients
The expression is in the form \( ay^2 + by + c \), where:
- \( a = 1 \) (coefficient of \( y^2 \))
- \( b = 5 \) (coefficient of \( y \))
- \( c = -24 \) (constant term)

### Step 2: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \( ac = 1 \times -24 = -24 \) and add up to \( b = 5 \).

### Step 3: List the factor pairs of \(-24\)
The factor pairs of \(-24\) are:
- \( (1, -24) \)
- \( (-1, 24) \)
- \( (2, -12) \)
- \( (-2, 12) \)
- \( (3, -8) \)
- \( (-3, 8) \)
- \( (4, -6) \)
- \( (-4, 6) \)

### Step 4: Check which pair adds up to \( 5 \)
Now, we check which of these pairs adds up to \( 5 \):
- \( 3 + (-8) = -5 \)
- \( -3 + 8 = 5 \) (This is the pair we need)

### Step 5: Rewrite the middle term using the found numbers
We can rewrite the expression \( y^2 + 5y - 24 \) as:
\[ y^2 - 3y + 8y - 24 \]

### Step 6: Group the terms
Now, we group the terms:
\[ (y^2 - 3y) + (8y - 24) \]

### Step 7: Factor out the common terms
Now, we factor out the common terms from each group:
\[ y(y - 3) + 8(y - 3) \]

### Step 8: Factor out the common binomial
Now, we can factor out the common binomial \( (y - 3) \):
\[ (y - 3)(y + 8) \]

### Final Answer
Thus, the factorised form of \( y^2 + 5y - 24 \) is:
\[ (y - 3)(y + 8) \]

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