Solve $\frac{1}{2}{x}^2+2x-3=0$ using the quadratic formula and simplify your answers.

Compute the discriminant of $x^2-8x+16=0$ and state how many real solutions the equation has.

Find the complex roots of $x^2+4x+13=0$ and express them in the simplest $a+bi$ form.

A projectile's height (in meters) after $t$ seconds is modeled by $h\left(t\right)=-5t^2+20t+15$. Use the quadratic formula to find when $h\left(t\right)=0$. Identify which (if any) is physically meaningful.

Given $x^2-5x=0$, which method is most efficient in solving it, and why?