Draw the graph of y=2x2-3x+0.5 (a parabola with a vertex at (¾,-⅝) and y-intercept at (0,½)). This can also been done on a graphing calculator (such as a Texas Instruments TI-83 Plus), which would probably also tell you the x intercepts, which are in fact the solutions (roots or zeroes) of 0=2x2-3x+0.5, because y=0 is the x-axis. The graph would give you approximate values for where the graph intercepts the x-axis; a graphing calculator would probably give you much more accurate values.
Approximate x-intercepts are (0.191,0) and (1.309), so the solutions are x=0.191 and 1.309. Either of these values satisfies the equation.