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

12
1 half

12, −4, 1
12, −4, 1

−4, 1
−4, 1

−4, 2, 3

Answer 1

To solve the equation \( x^2(2x-1) + 3x(2x-1) - 4(2x-1) = 0 \), we can first factor out the common term, which is \( (2x-1) \):

\[
(2x-1)(x^2 + 3x - 4) = 0
\]

Now we can set each factor equal to zero:

1. First, solve \( 2x - 1 = 0 \):
\[
2x = 1 \implies x = \frac{1}{2}
\]

2. Next, solve \( x^2 + 3x - 4 = 0 \). We can factor this quadratic:
\[
x^2 + 3x - 4 = (x + 4)(x - 1) = 0
\]
Setting each factor to zero gives:
\[
x + 4 = 0 \implies x = -4
\]
\[
x - 1 = 0 \implies x = 1
\]

Putting it all together, the complete set of solutions to the equation is:
\[
x = \frac{1}{2}, -4, 1
\]

Therefore, the correct response is:
\[
\frac{1}{2}, -4, 1
\]