To determine which value is in the solution set of the quadratic equation \(-2x^2 - 3x + 2 = 0\), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In this case, \(a = -2\), \(b = -3\), and \(c = 2\).
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Calculate the discriminant (\(b^2 - 4ac\)): \[ b^2 - 4ac = (-3)^2 - 4(-2)(2) = 9 + 16 = 25 \]
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Substitute the values into the quadratic formula: \[ x = \frac{-(-3) \pm \sqrt{25}}{2(-2)} = \frac{3 \pm 5}{-4} \]
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Calculate the two possible solutions:
- Using the plus sign: \[ x = \frac{3 + 5}{-4} = \frac{8}{-4} = -2 \]
- Using the minus sign: \[ x = \frac{3 - 5}{-4} = \frac{-2}{-4} = \frac{1}{2} \]
The solutions to the equation \(-2x^2 - 3x + 2 = 0\) are \(x = -2\) and \(x = \frac{1}{2}\).
Now let's check the options:
- −34
- Negative \( \frac{3}{4} \)
- −12
- Negative \( \frac{1}{2} \)
- 12
The only value from the options given that could potentially match is negative \( \frac{1}{2} \), but it is not an exact match since our solution yielded \( \frac{1}{2} \).
Therefore, based on the provided options, none of them are exact solutions based on calculations. However, if we expand the consideration for the potential approximation of values, the closest one could be negative \( \frac{1}{2} \), but it is indeed not a solution.
To summarize, neither negative \( \frac{3}{4} \), negative \( \frac{1}{2} \), or any of the available options are exactly in the solution set based on standard solution of the quadratic equation, as the actual solutions found do not appear among the responses.
If you are required to pick from given options, the option Negative \( \frac{1}{2} \) is the only candidate which implies a likelihood of a misunderstanding or typing errors in regard to original solutions listed.