Solve the (2x2 − x)2 − 9(2x2 − x) + 18 = 0

Sure, let's solve the equation step by step.

First, let's simplify the equation by substituting 2x^2 - x with y. So, the equation becomes:

y^2 - 9y + 18 = 0

This is a quadratic equation in the form of ax^2 + bx + c = 0. We can solve it by using the quadratic formula x = [-b ± sqrt(b^2 - 4ac)] / 2a.

Here, a = 1, b = -9, and c = 18.

So, the solutions for y are:

y1 = [9 + sqrt((-9)^2 - 4118)] / 21 y2 = [9 - sqrt((-9)^2 - 4118)] / 21

After calculating the above, we get:

y1 = 6 y2 = 3

Now, we substitute y back with 2x^2 - x to find the solutions for x:

2x^2 - x = 6 2x^2 - x - 6 = 0

Solving this quadratic equation, we get:

x1 = 2 x2 = -1.5

And for the second equation:

2x^2 - x = 3 2x^2 - x - 3 = 0

Solving this quadratic equation, we get:

x3 = 1.5 x4 = -1

So, the solutions for the original equation are x = 2, -1.5, 1.5, -1.