1. Stating the problem: Solve the quadratic equation involving complex terms: $$4x^2 - 7ix = 3$$ 2. Rewrite the equation in standard quadratic form: $$4x^2 - 7ix - 3 = 0$$ 3. Identify coefficients: $$a = 4, \quad b = -7i, \quad c = -3$$ 4. Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 5. Calculate the discriminant: $$b^2 - 4ac = (-7i)^2 - 4 \times 4 \times (-3) = (-7)^2 i^2 + 48 = 49(-1) + 48 = -49 + 48 = -1$$ 6. Substitute values into the formula: $$x = \frac{-(-7i) \pm \sqrt{-1}}{2 \times 4} = \frac{7i \pm i}{8}$$ 7. Simplify the expression: $$x = \frac{7i \pm i}{8} = \frac{i(7 \pm 1)}{8}$$ 8. Find the two solutions: - $$x_1 = \frac{i(7 + 1)}{8} = \frac{8i}{8} = i$$ - $$x_2 = \frac{i(7 - 1)}{8} = \frac{6i}{8} = \frac{3i}{4}$$ Final answer: $$x = i \quad \text{or} \quad x = \frac{3i}{4}$$