1. **State the problem:** Given the quadratic function $$y = 2x^2 - 0.4x + 2.5$$, we need to find how many roots it has using the discriminant and then find the roots using the quadratic formula if they exist. 2. **Recall the quadratic formula and discriminant:** For a quadratic equation $$ax^2 + bx + c = 0$$, the discriminant $$\Delta$$ is given by $$\Delta = b^2 - 4ac$$. - If $$\Delta > 0$$, there are 2 distinct real roots. - If $$\Delta = 0$$, there is 1 real root (a repeated root). - If $$\Delta < 0$$, there are no real roots (complex roots). The quadratic formula to find roots is: $$x = \frac{-b \pm \sqrt{\Delta}}{2a}$$ 3. **Identify coefficients:** $$a = 2, \quad b = -0.4, \quad c = 2.5$$ 4. **Calculate the discriminant:** $$\Delta = (-0.4)^2 - 4 \times 2 \times 2.5 = 0.16 - 20 = -19.84$$ 5. **Interpret the discriminant:** Since $$\Delta = -19.84 < 0$$, there are no real roots. The roots are complex. 6. **Calculate the roots using the quadratic formula:** $$x = \frac{-(-0.4) \pm \sqrt{-19.84}}{2 \times 2} = \frac{0.4 \pm \sqrt{-19.84}}{4}$$ Rewrite the square root of a negative number using imaginary unit $$i$$: $$\sqrt{-19.84} = i\sqrt{19.84}$$ So, $$x = \frac{0.4 \pm i\sqrt{19.84}}{4} = \frac{0.4}{4} \pm \frac{i\sqrt{19.84}}{4} = 0.1 \pm 0.25i\sqrt{3.1744}$$ Approximating $$\sqrt{19.84} \approx 4.455$$: $$x = 0.1 \pm \frac{4.455i}{4} = 0.1 \pm 1.11375i$$ **Final answer:** The quadratic has no real roots but two complex roots: $$x = 0.1 + 1.11375i$$ and $$x = 0.1 - 1.11375i$$.