To solve the equation \(-2x^2 + 3x - 1 = -2\), we first simplify it by moving all terms to one side:
\[ -2x^2 + 3x - 1 + 2 = 0 \]
This simplifies to:
\[ -2x^2 + 3x + 1 = 0 \]
For the quadratic formula, we generally use the form \(ax^2 + bx + c = 0\). Here, we have:
- \(a = -2\)
- \(b = 3\)
- \(c = 1\)
The quadratic formula is given by:
\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \]
First, we calculate the discriminant \(\Delta = b^2 - 4ac\):
\[ \Delta = 3^2 - 4 \cdot (-2) \cdot 1 = 9 + 8 = 17 \]
Now we can use the quadratic formula to find the solutions:
\[ x = \frac{{-3 \pm \sqrt{17}}}{2 \cdot -2} = \frac{{-3 \pm \sqrt{17}}{-4} \]
This simplifies to:
\[ x = \frac{{3 \mp \sqrt{17}}{4} \]
Now we calculate the two possible values for \(x\):
- Using the plus sign:
\[ x_1 = \frac{{3 + \sqrt{17}}}{4} \]
- Using the minus sign:
\[ x_2 = \frac{{3 - \sqrt{17}}}{4} \]
Next, we will calculate \(\sqrt{17}\):
\(\sqrt{17} \approx 4.123\)
Now substituting this value in:
- For \(x_1\):
\[ x_1 = \frac{{3 + 4.123}}{4} \approx \frac{7.123}{4} \approx 1.78075 \]
Rounding to the nearest hundredths gives:
\[ x_1 \approx 1.78 \]
- For \(x_2\):
\[ x_2 = \frac{{3 - 4.123}}{4} \approx \frac{-1.123}{4} \approx -0.28075 \]
Rounding to the nearest hundredths gives:
\[ x_2 \approx -0.28 \]
Thus, the final solutions, with the smaller solution first, are:
\[ \boxed{-0.28} \quad \text{and} \quad \boxed{1.78} \]
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