1. **State the problem:** Simplify or analyze the quadratic expression $x^2 - 5x + 25$. 2. **Recall the quadratic formula:** For any quadratic $ax^2 + bx + c$, the roots are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-5$, and $c=25$ here. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-5)^2 - 4 \times 1 \times 25 = 25 - 100 = -75$$ 4. **Interpret the discriminant:** Since $\Delta < 0$, the quadratic has no real roots; it has two complex conjugate roots. 5. **Find the roots:** $$x = \frac{-(-5) \pm \sqrt{-75}}{2 \times 1} = \frac{5 \pm \sqrt{-75}}{2} = \frac{5 \pm i\sqrt{75}}{2}$$ 6. **Simplify the square root:** $$\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}$$ 7. **Final roots:** $$x = \frac{5 \pm 5i\sqrt{3}}{2} = \frac{5}{2} \pm \frac{5\sqrt{3}}{2}i$$ **Answer:** The quadratic $x^2 - 5x + 25$ has no real roots and two complex roots: $$x = \frac{5}{2} \pm \frac{5\sqrt{3}}{2}i$$