\[ z = \frac{-(1 + i) \pm \sqrt{(1 + i)^2 - 4(1)(i)}}{2} \]

["Solving the Quadratic Equation ( z = \frac{-(1 + i) \pm \sqrt{(1 + i)^2 - 4i}}{2} ): A Complete Guide", "---", "Introduction", "Complex numbers often appear daunting at first, but solving quadratic equations involving them opens up powerful tools in mathematics, engineering, and physics. This article explores the quadratic equation:", "[\nz = \frac{-(1 + i) \pm \sqrt{(1 + i)^2 - 4(1)(i)}}{2}\n]", "We will break down each step, analyze the square root of complex expressions, simplify radicals, and provide a full derivation of the solutions — all optimized for search engines while remaining accessible to learners and professionals alike.", "---", "Step 1: Understand the Quadratic Formula in Complex Form", "The standard quadratic formula applies equally to complex coefficients:", "[\nz = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Here, ( a = 1 ), ( b = 1 + i ), and ( c = i ). Substituting these values gives:", "[\nz = \frac{-(1 + i) \pm \sqrt{(1 + i)^2 - 4 \cdot 1 \cdot i}}{2}\n]", "Our focus is on computing the discriminant:\n[\n\Delta = (1 + i)^2 - 4i\n]", "---", "Step 2: Compute the Discriminant ( \Delta = (1 + i)^2 - 4i )", "First, expand ( (1 + i)^2 ):", "[\n(1 + i)^2 = 1^2 + 2 \cdot 1 \cdot i + i^2 = 1 + 2i - 1 = 2i\n]", "Now subtract ( 4i ):", "[\n\Delta = 2i - 4i = -2i\n]", "So the discriminant simplifies to:", "[\n\Delta = -2i\n]", "---", "Step 3: Find ( \sqrt{-2i} ) — The Square Root of a Complex Number", "We seek complex numbers ( z = x + iy ) such that:", "[\nz^2 = -2i\n]", "Let’s set:", "[\n(x + iy)^2 = x^2 + 2xiy + (iy)^2 = x^2 - y^2 + 2xy i\n]", "Equate real and imaginary parts to ( -2i = 0 - 2i ):", "[\n\begin{cases}\nx^2 - y^2 = 0 \\n2xy = -2\n\end{cases}\n]", "From the first equation:\n[\nx^2 = y^2 \implies y = \pm x\n]", "Substitute into the second:\n- If ( y = x ), then ( 2x^2 = -2 \Rightarrow x^2 = -1 ), no real solution.\n- If ( y = -x ), then ( 2x(-x) = -2x^2 = -2 \Rightarrow x^2 = 1 \Rightarrow x = \pm 1 )", "Thus, solutions are:", "- ( x = 1, y = -1 \Rightarrow z = 1 - i )\n- ( x = -1, y = 1 \Rightarrow z = -1 + i )", "Therefore:", "[\n\sqrt{-2i} = 1 - i \quad \ ext{or} \quad -1 + i\n]", "Both are valid square roots; we use one appropriately. For consistency, we select ( \sqrt{-2i} = 1 - i ).", "---", "Step 4: Plug Back into Quadratic Formula", "Now substitute ( \sqrt{\Delta} = 1 - i ) into the original expression:", "[\nz = \frac{-(1 + i) \pm (1 - i)}{2}\n]", "Compute both solutions:", "First solution (( + ) sign):", "[\nz_1 = \frac{-(1 + i) + (1 - i)}{2} = \frac{-1 - i + 1 - i}{2} = \frac{-2i}{2} = -i\n]", "Second solution (( - ) sign):", "[\nz_2 = \frac{-(1 + i) - (1 - i)}{2} = \frac{-1 - i - 1 + i}{2} = \frac{-2}{2} = -1\n]", "---", "Step 5: Final Answer", "The roots of the equation:", "[\nz = \frac{-(1 + i) \pm \sqrt{(1 + i)^2 - 4i}}{2}\n]", "are:", "[\n\boxed{z_1 = -i \quad \ ext{and} \quad z_2 = -1}\n]", "---", "Conclusion", "Solving quadratic equations with complex coefficients is a meaningful exercise that combines algebraic manipulation and geometric insight. By carefully computing the discriminant and solving for the square roots in the complex plane, we found two exact solutions. This method is foundational in signal processing, control theory, and quantum mechanics.", "For anyone working with complex numbers, mastering such expressions strengthens both theoretical understanding and practical problem-solving skills. Optimize your learning by revisiting complex arithmetic, square roots of complex numbers, and the geometric interpretation of complex roots.", "---", "SEO Keywords:\ncomplex numbers, quadratic equation complex roots, solve ( z = \frac{-(1+i)\pm\sqrt{(1+i)^2 - 4i}}{2} ), discriminant in complex math, square root of (-2i), complex analysis tutorial, complex quadratic formula", "Meta Description:\nLearn how to solve ( z = \frac{-(1 + i) \pm \sqrt{(1 + i)^2 - 4i}}{2} ) with step-by-step calculation of the discriminant, square roots in complex numbers, and final roots explained clearly for students and professionals.", "---", "By combining clear explanation, precise mathematics, and strategic keyword placement, this article serves both educational and search visibility goals."]