1. **State the problem:** Simplify or analyze the quadratic expression $2x^2 - 5x + 2$. 2. **Recall the quadratic formula:** For any quadratic equation $ax^2 + bx + c = 0$, the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 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. **Find the roots:** $$x = \frac{-(-5) \pm \sqrt{9}}{2 \times 2} = \frac{5 \pm 3}{4}$$ 5. **Evaluate each root:** - For $+$ sign: $$x = \frac{5 + 3}{4} = \frac{8}{4} = 2$$ - For $-$ sign: $$x = \frac{5 - 3}{4} = \frac{2}{4} = \frac{1}{2}$$ 6. **Factor the quadratic using roots:** $$2x^2 - 5x + 2 = 2(x - 2)(x - \frac{1}{2})$$ 7. **Rewrite with integer coefficients:** $$2(x - 2)(x - \frac{1}{2}) = (x - 2)(2x - 1)$$ **Final answer:** The factorization of $2x^2 - 5x + 2$ is $$ (x - 2)(2x - 1) $$