Which one of the following could be a solution of the equation `x^2 - 7x - 18 = 0`?

Options

A. `-1`
B. `0`
C. `2`
D. `9`

Answer

To solve the quadratic equation \( x^2 - 7x - 18 = 0 \) and find its possible solutions, we can use the method of factorization. Here are the steps:

### Step 1: Identify the coefficients
The given equation is in the standard form \( ax^2 + bx + c = 0 \), where:
- \( a = 1 \)
- \( b = -7 \)
- \( c = -18 \)

### Step 2: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \( ac = 1 \times -18 = -18 \) and add to \( b = -7 \).

The numbers that satisfy these conditions are \( -9 \) and \( 2 \) because:
- \( -9 \times 2 = -18 \)
- \( -9 + 2 = -7 \)

### Step 3: Rewrite the equation using these numbers
We can rewrite the middle term of the equation using \( -9 \) and \( 2 \):
\[ x^2 - 9x + 2x - 18 = 0 \]

### Step 4: Factor by grouping
Now, we can group the terms:
\[ (x^2 - 9x) + (2x - 18) = 0 \]
Factoring out the common factors from each group:
\[ x(x - 9) + 2(x - 9) = 0 \]
Now, we can factor out \( (x - 9) \):
\[ (x - 9)(x + 2) = 0 \]

### Step 5: Set each factor to zero
Now, we can set each factor equal to zero to find the solutions:
1. \( x - 9 = 0 \) → \( x = 9 \)
2. \( x + 2 = 0 \) → \( x = -2 \)

### Conclusion
The possible solutions to the equation \( x^2 - 7x - 18 = 0 \) are \( x = 9 \) and \( x = -2 \).

### Answer
The answer is \( x = 9 \) or \( x = -2 \).

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