1. **State the problem:** Solve the quadratic equation $2x^2 - 3x + 4 = 0$ using the discriminant formula. 2. **Recall the quadratic formula and discriminant:** The quadratic formula to find roots of $ax^2 + bx + c = 0$ is $$x = \frac{-b \pm \sqrt{D}}{2a}$$ where the discriminant $D = b^2 - 4ac$ determines the nature of the roots. 3. **Identify coefficients:** Here, $a = 2$, $b = -3$, and $c = 4$. 4. **Calculate the discriminant:** $$D = (-3)^2 - 4 \times 2 \times 4 = 9 - 32 = -23$$ 5. **Interpret the discriminant:** Since $D < 0$, the equation has two complex conjugate roots. 6. **Find the roots:** $$x = \frac{-(-3) \pm \sqrt{-23}}{2 \times 2} = \frac{3 \pm \sqrt{-23}}{4} = \frac{3 \pm i\sqrt{23}}{4}$$ 7. **Final answer:** $$x = \frac{3}{4} \pm \frac{i\sqrt{23}}{4}$$ This means the solutions are complex numbers with real part $\frac{3}{4}$ and imaginary part $\frac{\sqrt{23}}{4}$.