To solve the equation \(-2x^2 + 3x - 1 = -2\), we first rearrange it into standard quadratic form \(ax^2 + bx + c = 0\).
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Add 2 to both sides: \[ -2x^2 + 3x - 1 + 2 = 0 \] This simplifies to: \[ -2x^2 + 3x + 1 = 0 \]
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Now, let's identify the coefficients \(a\), \(b\), and \(c\):
- \(a = -2\)
- \(b = 3\)
- \(c = 1\)
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We use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
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Calculate the discriminant \(b^2 - 4ac\): \[ b^2 - 4ac = 3^2 - 4(-2)(1) = 9 + 8 = 17 \]
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Now apply the values into the quadratic formula: \[ x = \frac{-3 \pm \sqrt{17}}{2(-2)} = \frac{-3 \pm \sqrt{17}}{-4} \]
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Splitting into two solutions: \[ x_1 = \frac{-3 + \sqrt{17}}{-4}, \quad x_2 = \frac{-3 - \sqrt{17}}{-4} \]
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Calculating \(\sqrt{17}\):
- \(\sqrt{17} \approx 4.1231\)
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Finding \(x_1\) and \(x_2\): \[ x_1 = \frac{-3 + 4.1231}{-4} = \frac{1.1231}{-4} \approx -0.2808 \implies -0.28 \text{ (rounded to hundredths)} \] \[ x_2 = \frac{-3 - 4.1231}{-4} = \frac{-7.1231}{-4} \approx 1.7808 \implies 1.78 \text{ (rounded to hundredths)} \]
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Final solutions: The two solutions are: \[ x \approx -0.28 \quad \text{or} \quad x \approx 1.78 \]
Thus, the final answer is: \[ x = -0.28 \quad \text{or} \quad 1.78 \]
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