39 8. \( -2 x^{2}+2 x=5 \)

To solve the equation \( -2x^{2} + 2x = 5 \), we first rearrange it into standard form by moving all terms to one side: \( -2x^{2} + 2x - 5 = 0 \). Next, we can multiply the entire equation by -1 to make calculations easier: \( 2x^{2} - 2x + 5 = 0 \). Now, we can apply the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), with \( a = 2, b = -2, \text{ and } c = -5 \). To find the discriminant, we calculate \( b^2 - 4ac = (-2)^{2} - 4(2)(-5) = 4 + 40 = 44 \). Since the discriminant is positive, we will have two real and distinct solutions. Plug these values into the quadratic formula, and we find the two solutions, which are approximately \( x = 2.5 \) and \( x = -1 \).