\[ (a + bi)^2 = a^2 - b^2 + 2abi = -2i \] - United Radiology

["# Solving ((a + bi)^2 = -2i): Step-by-Step Complex Algebra", "When working with complex numbers, squaring expressions like ((a + bi)^2) reveals fascinating mathematical insights. One common problem is solving equations of the form:", "[
\n(a + bi)^2 = -2i
\n]", "This equation appears simple at first glance but involves a deep dive into complex number arithmetic, conjugates, and polynomial solving. In this comprehensive guide, we will break down how to solve ((a + bi)^2 = -2i) step-by-step, explain key concepts, and explain why this matter important in engineering, physics, and higher mathematics.", "---", "## Understanding Complex Numbers and Squaring", "A complex number ( z = a + bi ) consists of a real part ( a ) and an imaginary part ( bi ), where ( i = \sqrt{-1} ). Squaring it involves applying the distributive law:", "[
\n(a + bi)^2 = a^2 + 2abi + (bi)^2 = a^2 + 2abi + b^2i^2
\n]", "Since ( i^2 = -1 ), this simplifies to:", "[
\na^2 - b^2 + 2abi
\n]", "So:", "[
\n(a + bi)^2 = (a^2 - b^2) + (2ab)i
\n]", "---", "## Step 1: Set Up the Equation", "Given:", "[
\n(a + bi)^2 = -2i
\n]", "Using the expanded form:", "[
\n(a^2 - b^2) + (2ab)i = 0 - 2i
\n]", "Now equate the real and imaginary parts:", "- Real part: ( a^2 - b^2 = 0 )
\n- Imaginary part: ( 2ab = -2 )", "We now solve this system of equations.", "---", "## Step 2: Solve the System of Equations", "From the real part:", "[
\na^2 - b^2 = 0 \Rightarrow a^2 = b^2 \Rightarrow b = \pm a
\n]", "Now substitute into the imaginary part:", "Case 1: ( b = a )
\nThen:", "[
\n2a \cdot a = -2 \Rightarrow 2a^2 = -2 \Rightarrow a^2 = -1
\n]", "This yields ( a^2 = -1 ), which has no real solutions — invalid for real ( a ).", "Case 2: ( b = -a )
\nNow:", "[
\n2a(-a) = -2 \Rightarrow -2a^2 = -2 \Rightarrow a^2 = 1 \Rightarrow a = \pm 1
\n]", "Using ( b = -a ), we get:", "- If ( a = 1 ), then ( b = -1 )
\n- If ( a = -1 ), then ( b = 1 )", "---", "## Step 3: Final Solutions", "Thus, the only real solutions are:", "[
\na + bi = 1 - i \quad \ ext{or} \quad -1 + i
\n]", "These are the two complex numbers whose squares equal (-2i).", "---", "## Verification", "Let’s verify ( (1 - i)^2 ):", "[
\n(1 - i)^2 = 1^2 - 2i + (i^2) = 1 - 2i - 1 = -2i
\n]", "Confirmed. Similarly for ((-1 + i)^2):", "[
\n(-1 + i)^2 = 1 - 2i + i^2 = 1 - 2i - 1 = -2i
\n]", "---", "## Why This Matters", "Solving ((a + bi)^2 = C) — especially with ( C ) a simple imaginary number like (-2i) — builds fluency with:", "- Complex conjugation and symmetry
\n- Square expansion and real-imaginary separation
\n- Root finding in the complex plane
\n- Applications in electrical engineering (impedance), quantum physics, signal processing, and control theory", "This problem exemplifies the power and elegance of algebra in the complex number system.", "---", "## Conclusion", "Solving ((a + bi)^2 = -2i) leads to two meaningful solutions: (1 - i) and (-1 + i). The key steps involve expanding the square, equating real and imaginary components, and solving the resulting equations systematically. This example not only reinforces core complex number techniques but also showcases how abstract algebra translates into practical problem-solving across science and engineering disciplines.", "---", "### Related Keywords for SEO Optimization:
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