Which of the following is a factor of 2 x squared minus x minus 6 equals 0?

Question

(1 point)
Responses

x minus 3
Image with alt text: x minus 3

x + 2
x + 2

2x + 3
2 x + 3

Answer 1

To determine which of the given options is a factor of the quadratic equation \( 2x^2 - x - 6 = 0 \), we can factor the quadratic expression.

We are looking for two numbers that multiply to \( 2 \times (-6) = -12 \) and add to \( -1 \) (the coefficient of the middle term, \( -x \)).

The numbers that meet these criteria are \( -4 \) and \( 3 \).

Now, we can rewrite the middle term:

\[ 2x^2 - 4x + 3x - 6 = 0 \]

Next, we factor by grouping:

\[ (2x^2 - 4x) + (3x - 6) = 0 \]

This simplifies to:

\[ 2x(x - 2) + 3(x - 2) = 0 \]

Now, we can factor out the common term \( (x - 2) \):

\[ (2x + 3)(x - 2) = 0 \]

Therefore, the factors of the quadratic are \( (2x + 3) \) and \( (x - 2) \).

Now we can check the options:

1. \( x - 3 \) (not a factor)
2. \( x + 2 \) (not a factor)
3. \( 2x + 3 \) (is a factor)

Thus, the correct response is:

2x + 3