To solve the equation
\[ -2x^2 + 3x - 1 = -2, \]
we first rearrange the equation into standard quadratic form by adding 2 to both sides:
\[ -2x^2 + 3x + 1 = 0. \]
Next, we identify the coefficients \(a\), \(b\), and \(c\):
\[ a = -2, \quad b = 3, \quad c = 1. \]
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \]
Now, we calculate the discriminant (\(b^2 - 4ac\)):
\[ b^2 - 4ac = 3^2 - 4(-2)(1) = 9 + 8 = 17. \]
Since the discriminant is positive, there will be two real solutions. Now we can substitute \(a\), \(b\), and the discriminant into the quadratic formula:
\[ x = \frac{-3 \pm \sqrt{17}}{2(-2)} = \frac{-3 \pm \sqrt{17}}{-4}. \]
This simplifies to:
\[ x = \frac{3 \mp \sqrt{17}}{4}. \]
Now we calculate the two solutions:
- For the positive case:
\[ x_1 = \frac{3 + \sqrt{17}}{4}, \]
- For the negative case:
\[ x_2 = \frac{3 - \sqrt{17}}{4}. \]
Next, we calculate \(\sqrt{17} \approx 4.123\):
- For \(x_1\):
\[ x_1 = \frac{3 + 4.123}{4} = \frac{7.123}{4} \approx 1.78075. \]
- For \(x_2\):
\[ x_2 = \frac{3 - 4.123}{4} = \frac{-1.123}{4} \approx -0.28075. \]
Now rounding both solutions to the nearest hundredths:
\[ x_1 \approx 1.78, \quad x_2 \approx -0.28. \]
Finally, we list the solutions in ascending order:
\[ \boxed{-0.28} \quad \text{and} \quad \boxed{1.78}. \]
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