`a^(2)+15a -54` =_______

Options

A. `(a+9)(x-6)`
B. `(a-18)(a+3)`
C. `(a+18)(a-3)`
D. `(a-27)(a+2)`

Answer

To factor the expression \( a^2 + 15a - 54 \), we can follow these steps:

### Step 1: Identify the coefficients
We have the quadratic expression in the form \( ax^2 + bx + c \), where:
- \( a = 1 \) (coefficient of \( a^2 \))
- \( b = 15 \) (coefficient of \( a \))
- \( c = -54 \) (constant term)

### Step 2: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers \( x \) and \( y \) such that:
- \( x \cdot y = ac = 1 \cdot (-54) = -54 \)
- \( x + y = b = 15 \)

### Step 3: List the factor pairs of \(-54\)
The factor pairs of \(-54\) are:
- \( (1, -54) \)
- \( (-1, 54) \)
- \( (2, -27) \)
- \( (-2, 27) \)
- \( (3, -18) \)
- \( (-3, 18) \)
- \( (6, -9) \)
- \( (-6, 9) \)

### Step 4: Check which pair adds up to \( 15 \)
Now, we check which of these pairs adds up to \( 15 \):
- \( 1 + (-54) = -53 \)
- \( -1 + 54 = 53 \)
- \( 2 + (-27) = -25 \)
- \( -2 + 27 = 25 \)
- \( 3 + (-18) = -15 \)
- \( -3 + 18 = 15 \) (This is our pair!)
- \( 6 + (-9) = -3 \)
- \( -6 + 9 = 3 \)

The pair we found is \( -3 \) and \( 18 \).

### Step 5: Rewrite the middle term using the found numbers
We can rewrite the expression as:
\[ a^2 + 18a - 3a - 54 \]

### Step 6: Factor by grouping
Now, we group the terms:
\[ (a^2 + 18a) + (-3a - 54) \]
Factoring out the common terms in each group:
\[ a(a + 18) - 3(a + 18) \]

### Step 7: Factor out the common binomial
Now we can factor out \( (a + 18) \):
\[ (a - 3)(a + 18) \]

### Final Answer
Thus, the factorization of \( a^2 + 15a - 54 \) is:
\[ (a - 3)(a + 18) \]