Solve 2𝑥2 + 2𝑥 − 2 = 3𝑥2 − 4𝑥

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

1. First, let's bring all terms to one side of the equation. We subtract 3x² from both sides and add 4x to both sides to get:

   2x² - 3x² + 2x + 4x - 2 = 0

2. Simplify the equation:

   -x² + 6x - 2 = 0

3. Now, we can solve for x using the quadratic formula, which is x = [-b ± sqrt(b² - 4ac)] / 2a. Here, a = -1, b = 6, and c = -2.

4. Substitute a, b, and c into the formula:

   x = [-6 ± sqrt((6)² - 4*(-1)(-2))] / 2(-1) x = [-6 ± sqrt(36 - 8)] / -2 x = [-6 ± sqrt(28)] / -2 x = [-6 ± 2*sqrt(7)] / -2

5. Simplify further to get the solutions:

   x = 3 ± sqrt(7)

So, the solutions to the equation 2x² + 2x - 2 = 3x² - 4x are x = 3 + sqrt(7) and x = 3 - sqrt(7).