1. **State the problem:** Solve the quadratic equation $$\frac{1}{2}x^2 - 5x + 2 = 0$$ using the quadratic formula. 2. **Recall the quadratic formula:** For an equation $$ax^2 + bx + c = 0$$, the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a$, $b$, and $c$ are coefficients. 3. **Identify coefficients:** Here, $a = \frac{1}{2}$, $b = -5$, and $c = 2$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-5)^2 - 4 \times \frac{1}{2} \times 2 = 25 - 4 = 21$$ Since $\Delta > 0$, there are two real solutions. 5. **Apply the quadratic formula:** $$x = \frac{-(-5) \pm \sqrt{21}}{2 \times \frac{1}{2}} = \frac{5 \pm \sqrt{21}}{1} = 5 \pm \sqrt{21}$$ 6. **Final answer:** $$x = 5 + \sqrt{21}, \quad x = 5 - \sqrt{21}$$ These are the simplified real solutions to the equation.