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?
1/2,-4,1
1/2
1/2,4,-1
-4,1
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.
To solve the equation, let's simplify it step by step:
1. Start with the given equation: x^2(2x−1)+3x(2x−1)−4(2x−1)=0
2. Notice that (2x−1) appears in every term. We can factor it out: (2x−1)(x^2+3x−4)=0
3. Now, we have a product of two factors that are equal to zero. So, we can set each factor equal to zero and solve for x separately:
a. (2x−1) = 0
Solving this equation, we find x = 1/2.
b. (x^2+3x−4) = 0
Factoring this quadratic equation, we have:
(x+4)(x−1) = 0
Solving this equation, we find x = -4, x = 1.
Therefore, the complete set of solutions to Janet's equation is x = 1/2, x = -4, and x = 1.
The correct option is 1/2, -4, 1.