To solve the equation \(-2x^2 + 3x - 1 = -2\) using the quadratic formula, we first need to rearrange it into standard form \(ax^2 + bx + c = 0\).
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Start by adding \(2\) to both sides:
\(-2x^2 + 3x - 1 + 2 = 0\)
\(-2x^2 + 3x + 1 = 0\)
Now we have the equation in standard form: \[ -2x^2 + 3x + 1 = 0 \] Here, \(a = -2\), \(b = 3\), and \(c = 1\).
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The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
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Calculate the discriminant \(b^2 - 4ac\):
\[ b^2 - 4ac = 3^2 - 4(-2)(1) = 9 + 8 = 17 \]
- Plug in the values into the quadratic formula:
\[ x = \frac{-3 \pm \sqrt{17}}{2(-2)} \] This simplifies to: \[ x = \frac{-3 \pm \sqrt{17}}{-4} \] Which can be rewritten as: \[ x = \frac{3 \mp \sqrt{17}}{4} \]
- Calculate the two potential solutions:
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For the minus sign: \[ x_1 = \frac{3 - \sqrt{17}}{4} \]
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For the plus sign: \[ x_2 = \frac{3 + \sqrt{17}}{4} \]
- Now, let's calculate these values. First, find \(\sqrt{17} \approx 4.123\):
- \(x_1 = \frac{3 - 4.123}{4} \approx \frac{-1.123}{4} \approx -0.28075\)
- \(x_2 = \frac{3 + 4.123}{4} \approx \frac{7.123}{4} \approx 1.78075\)
- Now, rounding to the nearest hundredths:
- \(x_1 \approx -0.28\)
- \(x_2 \approx 1.78\)
- Enter the smaller solution first: \[ x = -0.28 \text{ or } 1.78 \]
So the final answer is: \[ x = -0.28 \text{ or } 1.78 \]
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