To solve the equation \(0 = 2x^2 - 5x - 3\), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a = 2\), \(b = -5\), and \(c = -3\).
First, we will calculate the discriminant \(b^2 - 4ac\):
\[ b^2 = (-5)^2 = 25 \] \[ 4ac = 4 \cdot 2 \cdot (-3) = -24 \] \[ b^2 - 4ac = 25 - (-24) = 25 + 24 = 49 \]
Now we can substitute these values into the quadratic formula:
\[ x = \frac{-(-5) \pm \sqrt{49}}{2 \cdot 2} = \frac{5 \pm 7}{4} \]
Now, we will calculate the two possible values for \(x\):
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First, using the positive square root: \[ x = \frac{5 + 7}{4} = \frac{12}{4} = 3 \]
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Next, using the negative square root: \[ x = \frac{5 - 7}{4} = \frac{-2}{4} = -\frac{1}{2} \]
Therefore, the solutions to the equation \(0 = 2x^2 - 5x - 3\) are:
\[ x = 3 \quad \text{and} \quad x = -\frac{1}{2} \]
Among the provided responses, the correct answer is:
x equals negative \(\frac{1}{2}\), x equals 3.