To solve the equation \( 2x^2 = 6x - 5 \), we can first rearrange it into standard quadratic form by moving all terms to one side of the equation:
\[ 2x^2 - 6x + 5 = 0 \]
Next, we can use the quadratic formula to find the roots of the equation. The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 2 \), \( b = -6 \), and \( c = 5 \). We can first calculate the discriminant (\( b^2 - 4ac \)):
\[ b^2 - 4ac = (-6)^2 - 4 \cdot 2 \cdot 5 = 36 - 40 = -4 \]
Since the discriminant is negative, we will have complex solutions. Now we can apply the quadratic formula:
\[ x = \frac{-(-6) \pm \sqrt{-4}}{2 \cdot 2} = \frac{6 \pm \sqrt{-4}}{4} \]
Simplifying further:
\[ \sqrt{-4} = 2i \quad \text{(where \( i \) is the imaginary unit)} \]
Thus, we can express the solution as:
\[ x = \frac{6 \pm 2i}{4} \]
Simplifying this gives:
\[ x = \frac{6}{4} \pm \frac{2i}{4} = \frac{3}{2} \pm \frac{i}{2} \]
So, the two conjugate solution values are:
\[ x = \frac{3}{2} \pm \frac{i}{2} \]
Thus, the final answer is:
\[ \frac{3}{2} \pm \frac{i}{2} \]
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