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

−4
, 1
negative 4 , 1

12
Start Fraction 1 over 2 End Fraction

12
, −4
, 1
Start Fraction 1 over 2 End Fraction , negative 4 , 1

12
, 4, −1
Start Fraction 1 over 2 End Fraction , 4, negative 1

Answer 1

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

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

This gives us two equations to solve:

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

Starting with the first equation:

\[
2x - 1 = 0 \implies 2x = 1 \implies x = \frac{1}{2}
\]

Now, for the second equation \( x^2 + 3x - 4 = 0 \), we can factor it:

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

Setting each factor to zero gives us:

\[
x + 4 = 0 \implies x = -4
\]
\[
x - 1 = 0 \implies x = 1
\]

Thus, the complete set of solutions for Janet's equation is:

\[
x = \frac{1}{2}, \quad x = -4, \quad x = 1
\]

The correct response from the options you provided is:

**Start Fraction 1 over 2 End Fraction , negative 4 , 1** (or \( \frac{1}{2}, -4, 1 \)).