9. \( \frac{1}{2} x^{2}-\frac{3}{2} x+1=0 \)

To solve the quadratic equation \( \frac{1}{2} x^{2}-\frac{3}{2} x+1=0 \), you can first multiply through by 2 to eliminate the fractions. This gives you \( x^{2}-3x+2=0 \). You can then factor this to find the roots: \( (x-1)(x-2)=0 \), resulting in the solutions \( x=1 \) and \( x=2 \). Quadratic equations have been around for centuries, dating back to ancient Babylonians who tackled them with geometric methods long before algebra was a formalized discipline. They had an intuitive grasp of solving for areas of squares and rectangles, essentially stumbling upon today's roots and coefficients! In real life, quadratic equations pop up more often than you might think! Whether you're calculating trajectory in sports, optimizing profit in a business scenario, or even figuring out the dimensions of a garden plot, this mathematical tool is crucial. So next time you see a parabolic trajectory, remember, there’s some quadratic action happening behind the scenes!