\[ z^2 + (1 + i)z + i = 0 \]

["Solving the Complex Quadratic Equation: ( z^2 + (1 + i)z + i = 0 )", "In this comprehensive guide, we explore how to solve the quadratic equation ( z^2 + (1 + i)z + i = 0 ) in the complex plane. This equation combines real and imaginary coefficients, making it a fascinating example of solving polynomial equations with complex parameters. We’ll uncover the roots using the quadratic formula, interpret them geometrically, and explain their importance in fields like complex analysis, engineering, and control theory.", "---", "### Quick Overview", "The quadratic equation", "[\nz^2 + (1 + i)z + i = 0\n]", "has two complex roots, which we can find analytically. Solving such equations not only sharpens algebraic skills but also enhances understanding of complex numbers and their behavior — essential for advanced mathematics and applied sciences.", "---", "### Step-by-Step Solution Using the Quadratic Formula", "The standard quadratic formula works seamlessly for complex coefficients:", "[\nz = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "For our equation:\n( a = 1 ),\n( b = 1 + i ),\n( c = i ).", "Plug into the formula:", "[\nz = \frac{-(1 + i) \pm \sqrt{(1 + i)^2 - 4(1)(i)}}{2}\n]", "---", "### Step 1: Compute the Discriminant", "First, calculate ( b^2 - 4ac ):", "[\n(1 + i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i\n]", "[\n4ac = 4i\n]", "Thus,", "[\n\ ext{Discriminant} = (1 + i)^2 - 4i = 2i - 4i = -2i\n]", "So,", "[\nz = \frac{-(1 + i) \pm \sqrt{-2i}}{2}\n]", "---", "### Step 2: Simplify the Square Root of a Complex Number", "The tricky part is computing ( \sqrt{-2i} ). Let ( \sqrt{-2i} = x + yi ), where ( x, y \in \mathbb{R} ). Then:", "[\n(x + yi)^2 = -2i\n]", "Expanding:", "[\nx^2 - y^2 + 2xyi = -2i\n]", "Equating real and imaginary parts:", "[\nx^2 - y^2 = 0 \quad \ ext{(1)}\n]\n[\n2xy = -2 \quad \ ext{(2)}\n]", "From (1):\n( x^2 = y^2 \Rightarrow y = \pm x )", "Substitute into (2):\nIf ( y = x ):\n( 2x^2 = -2 ) → no real solution (LHS positive)", "If ( y = -x ):\n( 2x(-x) = -2 \Rightarrow -2x^2 = -2 \Rightarrow x^2 = 1 \Rightarrow x = \pm 1 )", "Thus, two solutions:", "- ( x = 1 \Rightarrow y = -1 \Rightarrow \sqrt{-2i} = 1 - i )\n- ( x = -1 \Rightarrow y = 1 \Rightarrow \sqrt{-2i} = -1 + i )", "Both are valid square roots since squaring either gives (-2i).", "For definiteness, choose ( \sqrt{-2i} = 1 - i ).", "---", "### Step 3: Compute the Roots", "Now substitute back:", "[\nz = \frac{-(1 + i) \pm (1 - i)}{2}\n]", "First root (( + ) sign):", "[\nz_1 = \frac{-1 - i + 1 - i}{2} = \frac{-2i}{2} = -i\n]", "Second root (( - ) sign):", "[\nz_2 = \frac{-1 - i - (1 - i)}{2} = \frac{-1 - i - 1 + i}{2} = \frac{-2}{2} = -1\n]", "---", "### Final Answer", "The solutions to the equation ( z^2 + (1 + i)z + i = 0 ) are:", "[\n\boxed{z = -1 \quad \ ext{and} \quad z = -i}\n]", "Both are precise, simplified complex roots satisfying the original quadratic.", "---", "### Geometric Interpretation and Significance", "These roots lie on the complex plane at points ( (-1, 0) ) and ( (0, -1) ) respectively. They form a symmetric pattern reflecting the influence of the complex coefficient ( (1+i) ). Understanding such roots aids in filtering signals, analyzing stability in control systems, and solving differential equations—applications critical in electrical engineering and physics.", "---", "### Why This Equation Matters", "- Complex Analysis: Demonstrates that every non-degenerate quadratic with complex coefficients has two complex (possibly equal) roots — the Fundamental Theorem of Algebra in the complex domain.\n- Engineering: Models linear systems where complex dynamics are crucial, such as AC circuits or vibration analysis.\n- Problem Solving: Reinforces techniques for solving equations with mixed real-imaginary terms, foundational for advanced mathematical modeling.", "---", "### Summary", "Solving ( z^2 + (1 + i)z + i = 0 ) involves applying the quadratic formula and carefully computing square roots of complex numbers. This guide walked through step-by-step, from discriminant calculation to final root identification, emphasizing both algebraic precision and conceptual clarity. Whether you're a student mastering complex numbers or a professional applying them in practice, this equation exemplifies how elegant mathematics underlies real-world innovation.", "---", "Keywords: complex quadratic equation, quadratic formula complex roots, solve ( z^2 + (1 + i)z + i = 0 ), complex analysis, impedance in electrical circuits, solving complex equations, quadratic with imaginary coefficients.", "---", "For further reading:\n- Quadratic equations in the complex plane\n- Applications of complex roots in engineering\n- Analytical solutions for second-degree polynomials with ( a, b, c \in \mathbb{C} )", "---", "Want more deep dives into complex equations and their real-world impact? Explore our collection of advanced mathematical guides tailored for students, engineers, and researchers."]