To solve the equation \(-2x^2 + 3x - 1 = -2\), we first rearrange it to standard form:
\[ -2x^2 + 3x - 1 + 2 = 0 \]
This simplifies to:
\[ -2x^2 + 3x + 1 = 0 \]
Next, we will apply the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our equation, \(a = -2\), \(b = 3\), and \(c = 1\).
Now, we calculate the discriminant \(b^2 - 4ac\):
\[ b^2 - 4ac = 3^2 - 4 \cdot (-2) \cdot 1 = 9 + 8 = 17 \]
Now, we can substitute \(a\), \(b\), and the discriminant into the quadratic formula:
\[ x = \frac{-3 \pm \sqrt{17}}{2 \cdot (-2)} = \frac{-3 \pm \sqrt{17}}{-4} \]
This can be simplified to:
\[ x = \frac{3 \mp \sqrt{17}}{4} \]
Now we calculate both solutions:
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For the positive root: \[ x_1 = \frac{3 + \sqrt{17}}{4} \]
Approximating \(\sqrt{17} \approx 4.123\): \[ x_1 \approx \frac{3 + 4.123}{4} \approx \frac{7.123}{4} \approx 1.78075 \] Rounding to the nearest hundredth, \(x_1 \approx 1.78\).
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For the negative root: \[ x_2 = \frac{3 - \sqrt{17}}{4} \]
Using \(\sqrt{17} \approx 4.123\): \[ x_2 \approx \frac{3 - 4.123}{4} \approx \frac{-1.123}{4} \approx -0.28075 \] Rounding to the nearest hundredth, \(x_2 \approx -0.28\).
So, the final answers rounded to the nearest hundredths are:
\[ x \approx -0.28 \text{ or } 1.78 \]
Thus, the final solutions:
\[ x = -0.28 \text{ or } 1.78 \]