Equations in Quadratic Form

Question

Solve: (2w−1)^2−1(2w−1)−2=0

Answer 1

Hi Genesis ,The equation you've provided is in quadratic form. To solve it " (2w−1)^2−1(2w−1)−2=0" let's make a substitution to simplify the equation:

Let y = 2w - 1.

Now, the equation becomes:

y^2 - y - 2 = 0.

This is a standard quadratic equation. We can solve it using the quadratic formula:

y = (-b ± √(b^2 - 4ac)) / (2a),

where a = 1, b = -1, and c = -2.

y = (-(-1) ± √((-1)^2 - 4 * 1 * (-2))) / (2 * 1)

y = (1 ± √(1 + 8)) / 2

y = (1 ± √9) / 2

Now, there are two possible solutions:

1. y = (1 + 3) / 2 = 4/2 = 2

2. y = (1 - 3) / 2 = -2/2 = -1

Now, we need to convert these solutions back to w:

1. For y = 2:

2w - 1 = 2

2w = 2 + 1

2w = 3

w = 3/2

2. For y = -1:

2w - 1 = -1

2w = -1 + 1

2w = 0

w = 0

So, the solutions for the original equation (2w - 1)^2 - 1(2w - 1) - 2 = 0 are:

w = 3/2 and w = 0.

I hope this will help. I am happy to tutor you on any other questions you may have; please feel free to send me a message!