To solve the equation \(-2x^2 + 3x - 1 = -2\), we first rearrange it into standard form, which takes the form \(ax^2 + bx + c = 0\).
Starting with the original equation:
\[ -2x^2 + 3x - 1 = -2 \]
Add \(2\) to both sides to get:
\[ -2x^2 + 3x - 1 + 2 = 0 \]
This simplifies to:
\[ -2x^2 + 3x + 1 = 0 \]
To put it in standard form, we can multiply the entire equation by \(-1\):
\[ 2x^2 - 3x - 1 = 0 \]
Now, we will use the quadratic formula, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In this case, the coefficients are:
- \(a = 2\)
- \(b = -3\)
- \(c = -1\)
Next, we calculate the discriminant (\(b^2 - 4ac\)):
\[ b^2 - 4ac = (-3)^2 - 4(2)(-1) = 9 + 8 = 17 \]
Now substitute the values back into the quadratic formula:
\[ x = \frac{-(-3) \pm \sqrt{17}}{2(2)} = \frac{3 \pm \sqrt{17}}{4} \]
Next, we need to calculate the two possible values of \(x\):
- For the positive root:
\[ x_1 = \frac{3 + \sqrt{17}}{4} \approx \frac{3 + 4.1231}{4} \approx \frac{7.1231}{4} \approx 1.7808 \]
- For the negative root:
\[ x_2 = \frac{3 - \sqrt{17}}{4} \approx \frac{3 - 4.1231}{4} \approx \frac{-1.1231}{4} \approx -0.2808 \]
Rounding the solutions to the nearest hundredths, we get:
\[ x_1 \approx 1.78 \quad \text{and} \quad x_2 \approx -0.28 \]
Thus, the solutions to the equation are:
\[ \boxed{1.78} \quad \text{and} \quad \boxed{-0.28} \]