To determine whether each of the given values is a solution to the quadratic equation \(-2x^2 - 3x + 2 = 0\), we will substitute each \(x\) value into the equation and check if it results in zero.
Here's how it's done for the given quadratic equation:
- Substitute values into the equation.
- Check if the equation equals zero.
We will evaluate the equation for the given value \(x = \frac{1}{2}\).
Step 1: Substitute \(x = \frac{1}{2}\) into the equation.
\[ -2\left(\frac{1}{2}\right)^2 - 3\left(\frac{1}{2}\right) + 2 = 0 \]
Calculating term by term:
\[ = -2\left(\frac{1}{4}\right) - \frac{3}{2} + 2 \] \[ = -\frac{2}{4} - \frac{3}{2} + 2 \] \[ = -\frac{1}{2} - \frac{3}{2} + 2 \]
Now convert \(2\) into halves:
\[ = -\frac{1}{2} - \frac{3}{2} + \frac{4}{2} \] \[ = -\frac{1 + 3 - 4}{2} \] \[ = -\frac{0}{2} = 0 \]
So, \(x = \frac{1}{2}\) is indeed a solution to the equation \(-2x^2 - 3x + 2 = 0\).
Since you've stated that only \(x = \frac{1}{2}\) is in the solution set, we can conclude that it is the only value that satisfies the equation.
You would, however, need to check other proposed values (not just \(x = \frac{1}{2}\)) if you want to complete the table based on additional values provided. If you have those values, please provide them for evaluation.
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