1. Stating the problem: Solve the quadratic equation $2x^2 + 5x + 2 = 0$. 2. Formula used: The quadratic formula for $ax^2 + bx + c = 0$ is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=2$, $b=5$, and $c=2$. 3. Calculate the discriminant: $$\Delta = b^2 - 4ac = 5^2 - 4 \times 2 \times 2 = 25 - 16 = 9$$ 4. Since $\Delta > 0$, there are two real roots. 5. Apply the quadratic formula: $$x = \frac{-5 \pm \sqrt{9}}{2 \times 2} = \frac{-5 \pm 3}{4}$$ 6. Calculate each root: - For $+$ sign: $$x = \frac{-5 + 3}{4} = \frac{-2}{4} = -\frac{1}{2}$$ - For $-$ sign: $$x = \frac{-5 - 3}{4} = \frac{-8}{4} = -2$$ 7. Final answer: The solutions are $x = -\frac{1}{2}$ and $x = -2$.