The factors of the expression `x^(2)-2x-63` are __________.

Options

A. `x-7,x-9`
B. `x+7,x+9`
C. `x+7,x-9`
D. `x-7,x+9`

Answer

To factor the expression \( x^2 - 2x - 63 \), we will follow these steps:

### Step 1: Identify the coefficients
The expression is in the standard form \( ax^2 + bx + c \), where:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = -2 \) (coefficient of \( x \))
- \( c = -63 \) (constant term)

### Step 2: Find two numbers that multiply to \( ac \) and add to \( b \)
Here, \( ac = 1 \times (-63) = -63 \) and \( b = -2 \). We need to find two numbers that multiply to \(-63\) and add to \(-2\).

The pairs of factors of \(-63\) are:
- \( 1 \) and \(-63\)
- \(-1\) and \(63\)
- \( 3 \) and \(-21\)
- \(-3\) and \(21\)
- \( 7 \) and \(-9\)
- \(-7\) and \(9\)

Among these, the pair \( 7 \) and \(-9\) adds up to \(-2\) (i.e., \( 7 + (-9) = -2 \)).

### Step 3: Rewrite the middle term using the two numbers
We can rewrite the expression \( x^2 - 2x - 63 \) as:
\[ x^2 + 7x - 9x - 63 \]

### Step 4: Factor by grouping
Now, we group the terms:
\[ (x^2 + 7x) + (-9x - 63) \]

Factoring out the common factors in each group:
\[ x(x + 7) - 9(x + 7) \]

### Step 5: Factor out the common binomial
Now we can factor out the common binomial \( (x + 7) \):
\[ (x + 7)(x - 9) \]

### Final Answer
The factors of the expression \( x^2 - 2x - 63 \) are:
\[ (x + 7)(x - 9) \]

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