eta^2 = (1 - i)^2 = 1 - 2i + i^2 = 1 - 2i - 1 = -2i

["Understanding β² = (1 - i)²: Solving Quadratic Expressions in the Complex Plane", "In mathematics, particularly in algebra and complex analysis, working with imaginary numbers opens doors to deeper insights into equations and transformations. One intriguing expression is ( \beta^2 = (1 - i)^2 ), where ( i ) is the imaginary unit defined by ( i^2 = -1 ). This article explores how to simplify this expression, revealing a key result in complex number computations.", "---", "### What is ( \beta^2 = (1 - i)^2 )?", "The expression ( \beta^2 = (1 - i)^2 ) involves squaring a complex binomial. Squaring a binomial follows the distributive property:", "[\n(1 - i)^2 = (1 - i)(1 - i)\n]", "Applying the FOIL method (First, Outer, Inner, Last):", "[\n(1 - i)(1 - i) = 1 \cdot 1 + 1 \cdot (-i) + (-i) \cdot 1 + (-i) \cdot (-i)\n]", "Calculating term by term:", "- (1 \cdot 1 = 1)\n- (1 \cdot (-i) = -i)\n- (-i \cdot 1 = -i)\n- (-i \cdot (-i) = i^2 = -1) (since ( i^2 = -1 ))", "Adding them all together:", "[\n1 - i - i - 1 = -2i\n]", "Thus,", "[\n\beta^2 = (1 - i)^2 = -2i\n]", "---", "### Interpreting ( \beta^2 = -2i ): What Does It Mean?", "The solution ( \beta^2 = -2i ) implies that both ( \beta = \sqrt{-2i} ) and its negative represent solutions in the complex plane. Understanding this requires finding complex square roots—an important concept in algebra and engineering.", "To find ( \beta ), suppose ( \beta = a + bi ), where ( a ) and ( b ) are real numbers. Then:", "[\n\beta^2 = (a + bi)^2 = a^2 + 2abi + (bi)^2 = a^2 - b^2 + 2abi\n]", "We want this to equal ( -2i ), which has real part 0 and imaginary part -2. Matching real and imaginary components:", "- Real part: ( a^2 - b^2 = 0 )\n- Imaginary part: ( 2ab = -2 )", "From ( a^2 = b^2 ), we get ( a = b ) or ( a = -b ).", "Case 1: ( a = b )\nThen ( 2ab = 2a^2 = -2 \Rightarrow a^2 = -1 ), which has no real solution.", "Case 2: ( a = -b )\nThen ( 2ab = 2a(-a) = -2a^2 = -2 \Rightarrow a^2 = 1 \Rightarrow a = 1 ) or ( a = -1 )", "So, ( a = 1 \Rightarrow b = -1 ) or ( a = -1 \Rightarrow b = 1 )", "Hence, the two complex solutions are:", "[\n\beta = 1 - i \quad \ ext{and} \quad \beta = -1 + i\n]", "These are precisely ( \beta = \pm \sqrt{-2i} ), confirming our earlier result.", "---", "### Why Does ( (1 - i)^2 = -2i ) Matter?", "1. Complex Square Roots: It illustrates how real operations yield complex results, expanding the domain beyond real numbers.\n2. Polar Form & De Moivre’s Theorem: Computing squares in rectangular form supports exploring polar representations, where magnitude and argument aid in power calculations.\n3. Minimal Polynomial: ( \beta^2 + 2i = 0 ) defines a quadratic relationship central in linear algebra and field extensions.\n4. Applications: Used in electrical engineering, signal processing, quantum mechanics, and control theory for modeling oscillations and oscillations in phasor domains.", "---", "### Summary", "The equation ( \beta^2 = (1 - i)^2 = -2i ) simplifies neatly using basic binomial expansion in complex arithmetic. This yields a purely imaginary result, demonstrating how squaring complex numbers generates rich structure beyond real-line computations. Solving ( \beta^2 = -2i ) requires identifying complex numbers whose square equals (-2i), leading to solutions involving both real and imaginary components—revealing the multifaceted nature of complex algebra.", "Whether you're solving equations, analyzing waveforms, or exploring rotations in the plane, mastering such identities is essential for navigating the complex plane with confidence.", "---", "Keywords: ( \beta^2 = (1 - i)^2 ), complex numbers, imaginary unit ( i ), solving quadratic expressions, complex square roots, ( 1 - i ) squared, ( -2i ), algebra with complex numbers, complex arithmetic."]