Let S be the set of all complex numbers z satisfying the equation 2|z| - z = 7 - 4i . The sum of the real parts of all elements in S is

Step-by-step solution

Let z = x + iy , where x, y R . Substitute z into the given equation: 2|x + iy| - (x + iy) = 7 - 4i 2 x^2 + y^2 - x - iy = 7 - 4i Equating the imaginary parts: -y = -4 y = 4 Equating the real parts: 2 x^2 + 16 - x = 7 2 x^2 + 16 = x + 7 Squaring both sides, we get: 4(x^2 + 16) = (x + 7)^2 4x^2 + 64 = x^2 + 14x + 49 3x^2 - 14x + 15 = 0 Factorising the quadratic equation: 3x^2 - 9x - 5x + 15 = 0 3x(x - 3) - 5(x - 3) = 0 (3x - 5)(x - 3) = 0 x = 5 3 or x = 3 We must verify if these roots satisfy the condition x + 7 0 (