1. **State the problem:** Solve the quadratic equation $2x^2 + 3x - 2 = 0$ for $x$. 2. **Formula used:** The quadratic formula is given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a$, $b$, and $c$ are coefficients from the quadratic equation $ax^2 + bx + c = 0$. 3. **Identify coefficients:** Here, $a = 2$, $b = 3$, and $c = -2$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 3^2 - 4 \times 2 \times (-2) = 9 + 16 = 25$$ 5. **Calculate the roots:** $$x = \frac{-3 \pm \sqrt{25}}{2 \times 2} = \frac{-3 \pm 5}{4}$$ 6. **Find each solution:** - For $+$ sign: $$x = \frac{-3 + 5}{4} = \frac{2}{4} = 0.5$$ - For $-$ sign: $$x = \frac{-3 - 5}{4} = \frac{-8}{4} = -2$$ **Final answer:** The solutions are $x = 0.5$ and $x = -2$.