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

Question

1/2,-4,1

1/2

1/2,4,-1

-4,1

Answer 1

To solve the equation, we can simplify the equation and solve for x. Simplifying the left side of the equation, we get:

x^2(2x−1)+3x(2x−1)−4(2x−1) = 0
2x^3 - x^2 + 6x^2 - 3x - 8x + 4 = 0
2x^3 + 5x^2 - 11x + 4 = 0

To find the solutions, we can try different values of x and see if they make the equation true. We can start by trying x = 1/2:

2(1/2)^3 + 5(1/2)^2 - 11(1/2) + 4 = 0
1/4 + 5/4 - 11/2 + 4 = 0
(1 + 5 - 22 + 16)/4 = 0
0/4 = 0

Since the equation is true for x = 1/2, we have found one solution.

Next, we can try x = -4:

2(-4)^3 + 5(-4)^2 - 11(-4) + 4 = 0
-128 + 80 + 44 + 4 = 0
0 = 0

Again, the equation is true for x = -4, so we have found another solution.

Finally, we can try x = 1:

2(1)^3 + 5(1)^2 - 11(1) + 4 = 0
2 + 5 - 11 + 4 = 0
0 = 0

The equation is also true for x = 1, so we have found the third solution.

Therefore, a complete set of solutions to Janet's equation is 1/2, -4, and 1.