This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Expand the term (2+4i)^2. Using the formula (a+b)^2 = a^2 + 2ab + b^2: (2+4i)^2 = 2^2 + 2(2)(4i) + (4i)^2 = 4 + 16i + 16i^2 Since i^2 = -1: = 4 + 16i + 16(-1) = 4 + 16i - 16 = -12 + 16i Step 2: Substitute this back into the original equation. The given equation is (2+4i)^2 - 2(x+iy) = x+iy. Substitute the expanded term: (-12 + 16i) - 2(x+iy) = x+iy Step 3: Distribute the -2 on the left side. -12 + 16i - 2x - 2iy = x+iy Step 4: Rearrange the terms to group real and imaginary parts on the left side. (-12 - 2x) + (16 - 2y)i = x+iy Step 5: Equate the real parts and the imaginary parts from both sides of the equation. For the real parts: -12 - 2x = x (1) For the imaginary parts: 16 - 2y = y (2) Step 6: Solve the system of equations for x and y. From equation (1): -12 - 2x = x -12 = x + 2x -12 = 3x x = (-12)/(3) x = -4 From equation (2): 16 - 2y = y 16 = y + 2y 16 = 3y y = (16)/(3) Step 7: Compare the calculated values with the given options. The values are x=-4 and y=(16)/(3). Looking at the options: A. -4, (16)/(3) B. -4, 5 C. 4, -3 D. 2, 16 The calculated values match option A. The final answer is A. x=-4, y=(16)/(3).