Janet is trying to solve an equation and gets stuck at x2(2x − 1)+3x(2x − 1)−4(2x−1) =0.

Question

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.