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'll use the quadratic formula, which states:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our equation, the coefficients are:
- \(a = -2\)
- \(b = 3\)
- \(c = 1\)
Calculating the discriminant \(b^2 - 4ac\):
\[ b^2 - 4ac = 3^2 - 4(-2)(1) = 9 + 8 = 17 \]
Now we can substitute back into the quadratic formula:
\[ x = \frac{-3 \pm \sqrt{17}}{2 \times -2} = \frac{-3 \pm \sqrt{17}}{-4} \]
This simplifies to:
\[ x = \frac{3 \mp \sqrt{17}}{4} \]
Now we calculate the two possible values of \(x\):
- \(x_1 = \frac{3 + \sqrt{17}}{4}\)
- \(x_2 = \frac{3 - \sqrt{17}}{4}\)
Calculating \(x_1\) and \(x_2\):
First, we find \(\sqrt{17} \approx 4.123\).
Calculating \(x_1\):
\[ x_1 = \frac{3 + 4.123}{4} = \frac{7.123}{4} \approx 1.78075 \]
Calculating \(x_2\):
\[ x_2 = \frac{3 - 4.123}{4} = \frac{-1.123}{4} \approx -0.28075 \]
Rounding to the nearest hundredths:
- \(x_1 \approx 1.78\)
- \(x_2 \approx -0.28\)
Thus, the solutions to the equation are:
\[ x \approx -0.28 \quad \text{or} \quad x \approx 1.78 \]
So, we write the final answer as:
\[ x = -0.28 \quad \text{or} \quad x = 1.78 \]
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