1. **State the problem:** We are given the quadratic expression $2x^2 - 5x + 2$ and want to analyze it. 2. **Formula and rules:** A quadratic expression is generally written as $ax^2 + bx + c$. Here, $a=2$, $b=-5$, and $c=2$. 3. **Find the roots using the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-5)^2 - 4 \times 2 \times 2 = 25 - 16 = 9$$ 5. **Calculate the roots:** $$x = \frac{-(-5) \pm \sqrt{9}}{2 \times 2} = \frac{5 \pm 3}{4}$$ 6. **Evaluate each root:** - For $+$ sign: $x = \frac{5 + 3}{4} = \frac{8}{4} = 2$ - For $-$ sign: $x = \frac{5 - 3}{4} = \frac{2}{4} = 0.5$ 7. **Vertex of the parabola:** The vertex $x$-coordinate is given by $x = -\frac{b}{2a} = -\frac{-5}{2 \times 2} = \frac{5}{4} = 1.25$ 8. **Calculate the vertex $y$-coordinate:** $$y = 2(1.25)^2 - 5(1.25) + 2 = 2(1.5625) - 6.25 + 2 = 3.125 - 6.25 + 2 = -1.125$$ 9. **Summary:** - Roots: $x=2$ and $x=0.5$ - Vertex: $(1.25, -1.125)$ - Since $a=2 > 0$, the parabola opens upwards. This completes the analysis of the quadratic $2x^2 - 5x + 2$.