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

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

−4
, 1
negative 4 , 1

12
Start Fraction 1 over 2 End Fraction

12
, −4
, 1

Answer 1

To solve Janet's equation \( x^2(2x−1) + 3x(2x−1) − 4(2x−1) = 0 \), we can factor out \( (2x - 1) \):

1. Rewrite the equation as: \[ (2x - 1)(x^2 + 3x - 4) = 0 \]

2. This gives us two equations to solve:

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

3. Solving \( 2x - 1 = 0 \): \[ 2x = 1 \implies x = \frac{1}{2} \]

4. Now, we solve the quadratic equation \( x^2 + 3x - 4 = 0 \) using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-3 \pm \sqrt{3^2 - (4)(1)(-4)}}{2(1)} = \frac{-3 \pm \sqrt{9 + 16}}{2} = \frac{-3 \pm \sqrt{25}}{2} = \frac{-3 \pm 5}{2} \]

5. This gives us two solutions: \[ x = \frac{2}{2} = 1 \quad \text{and} \quad x = \frac{-8}{2} = -4 \]

Therefore, the complete set of solutions is: \[ \frac{1}{2}, 1, -4 \]

Among the provided options, the correct answer is:

\[ \frac{1}{2}, 1, -4 \]