Which of the following is the factored form of `2x^2-3x-9`?

Options

A. `2x-3)(x-9)`
B. `(2x-9)(x+3)`
C. `(3x+2)(x-3)`
D. `(2x+3)(x-3)`

Answer

To factor the quadratic expression \(2x^2 - 3x - 9\), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic expression is in the form \(ax^2 + bx + c\), where:
- \(a = 2\)
- \(b = -3\)
- \(c = -9\)

### Step 2: Calculate the product \(ac\)
We need to calculate \(ac\):
\[ ac = 2 \times (-9) = -18 \]

### Step 3: Find two numbers that multiply to \(ac\) and add to \(b\)
We are looking for two numbers \(m\) and \(n\) such that:
- \(m \cdot n = -18\) (the product)
- \(m + n = -3\) (the sum)

After testing different pairs, we find:
- \(m = 3\) and \(n = -6\)

### Step 4: Rewrite the middle term
We can rewrite the expression \(2x^2 - 3x - 9\) using the numbers we found:
\[ 2x^2 + 3x - 6x - 9 \]

### Step 5: Group the terms
Now, we will group the terms:
\[ (2x^2 + 3x) + (-6x - 9) \]

### Step 6: Factor out the common terms
Now, we factor out the common factors from each group:
\[ x(2x + 3) - 3(2x + 3) \]

### Step 7: Factor by grouping
Now, we can factor out the common binomial factor \((2x + 3)\):
\[ (2x + 3)(x - 3) \]

### Step 8: Write the final factored form
Thus, the factored form of the expression \(2x^2 - 3x - 9\) is:
\[ (2x + 3)(x - 3) \]

### Final Answer
The factored form of \(2x^2 - 3x - 9\) is \((2x + 3)(x - 3)\).

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