Question

Janet is trying to solve an equation and gets stuck at x2(2x − 1)+3x(2x − 1)−4(2x−1) =0.
Which of the following is a complete set of solutions to Janet’s equation?(1 point)
Responses

1/2, -4, 1
1/2, -4, 1

1/2, 4, -1
1/2, 4, -1

1/2
1/2

-4,1

Answer 1

To solve the equation \( x^2(2x - 1) + 3x(2x - 1) - 4(2x - 1) = 0 \), we can factor by recognizing that \( (2x - 1) \) is a common factor in all terms:

1. Factor out \( (2x - 1) \):
\[
(2x - 1)(x^2 + 3x - 4) = 0
\]

2. Set each factor to zero:
- For \( 2x - 1 = 0 \):
\[
2x = 1 \implies x = \frac{1}{2}
\]

- For \( x^2 + 3x - 4 = 0 \), we can factor this quadratic:
\[
x^2 + 4x - x - 4 = 0 \implies (x + 4)(x - 1) = 0
\]
This gives us the solutions:
\[
x + 4 = 0 \implies x = -4
\]
\[
x - 1 = 0 \implies x = 1
\]

3. Thus, the complete set of solutions to Janet's equation is:
\[
x = \frac{1}{2}, -4, 1
\]

The correct response is:
**1/2, -4, 1**.