To determine which of the given values \(-\frac{3}{4}\), \(-\frac{1}{2}\), or \(\frac{1}{2}\) is a solution to the equation \(-2x^2 - 3x + 2 = 0\), we will substitute each value into the equation and check if the equation holds true (i.e., if the left-hand side equals zero).
Step 1: Substitute each value into the equation
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Testing \(x = -\frac{3}{4}\): \[ -2\left(-\frac{3}{4}\right)^2 - 3\left(-\frac{3}{4}\right) + 2 \] Calculate \((-3/4)^2\): \[ -2\left(\frac{9}{16}\right) + \frac{9}{4} + 2 \] Simplify: \[ -\frac{18}{16} + \frac{36}{16} + \frac{32}{16} = -\frac{18}{16} + \frac{68}{16} = \frac{50}{16} = \frac{25}{8} \neq 0 \]
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Testing \(x = -\frac{1}{2}\): \[ -2\left(-\frac{1}{2}\right)^2 - 3\left(-\frac{1}{2}\right) + 2 \] Calculate \((-1/2)^2\): \[ -2\left(\frac{1}{4}\right) + \frac{3}{2} + 2 \] Simplify: \[ -\frac{1}{2} + \frac{3}{2} + \frac{4}{2} = -\frac{1}{2} + \frac{7}{2} = \frac{6}{2} = 3 \neq 0 \]
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Testing \(x = \frac{1}{2}\): \[ -2\left(\frac{1}{2}\right)^2 - 3\left(\frac{1}{2}\right) + 2 \] Calculate \((1/2)^2\): \[ -2\left(\frac{1}{4}\right) - \frac{3}{2} + 2 \] Simplify: \[ -\frac{1}{2} - \frac{3}{2} + 2 = -\frac{4}{2} + 2 = -2 + 2 = 0 \]
Conclusion
After testing all three values, we find that:
- \(x = -\frac{3}{4}\) does not satisfy the equation.
- \(x = -\frac{1}{2}\) does not satisfy the equation.
- \(x = \frac{1}{2}\) satisfies the equation.
Thus, the solution is \(\frac{1}{2}\).