To solve the equation \( x^2 - 2x + 2 - 3x - x = 0 \), we first simplify it:
- Combine like terms: \[ x^2 - 2x - 3x - x + 2 = 0 \implies x^2 - 6x + 2 = 0 \]
Now we can solve the quadratic equation \( x^2 - 6x + 2 = 0 \) using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 1 \), \( b = -6 \), and \( c = 2 \).
-
Calculate the discriminant: \[ b^2 - 4ac = (-6)^2 - 4 \cdot 1 \cdot 2 = 36 - 8 = 28 \]
-
Now plug the values into the quadratic formula: \[ x = \frac{6 \pm \sqrt{28}}{2 \cdot 1} = \frac{6 \pm 2\sqrt{7}}{2} \]
-
Simplify: \[ x = 3 \pm \sqrt{7} \]
Thus, the solutions to the equation are: \[ x = 3 + \sqrt{7} \quad \text{and} \quad x = 3 - \sqrt{7} \]