а) z* = -z·i
z = x + iy
x - iy = -(x + iy)·i
x - iy = -ix + y
x + ix = y + iy
x·(1 + i) = y·(1 + i)
y = x
z = x + ix, x ∈ R
б) 2·|z| - 8z + 1 + 2i = 0
z = x + iy
2√(x² + y²) - 8·(x + iy) + 1 + 2i = 0
2√(x² + y²) - 8x - i8y + 1 + 2i = 0
2√(x² + y²) = (8x - 1) + i(8y - 2)
2√(x² + y²) = 8x - 1
8y - 2 = 0
y = 1/4
2√(x² + (1/4)²) = 8x - 1
4(x² + 1/16) = 64x² - 16x + 1
8x - 1 ≥ 1/2
4x² + 1/4 = 64x² - 16x + 1
8x ≥ 3/2
60x² - 16x + 3/4 = 0
x ≥ 3/16
240x² - 64x + 3 = 0
D = 64² - 4·240·3 = 1216
x = (64 (+/-) √1216)/480 = (64 (+/-) 8√19)/480 = (8 (+/-) √19)/60
x = 2/15 (+/-) √19/60
x ≥ 3/16
x = 2/15 + √19/60
z = 2/15 + √19/60 + i/4
а) z* = -z·i
z = x + iy
x - iy = -(x + iy)·i
x - iy = -ix + y
x + ix = y + iy
x·(1 + i) = y·(1 + i)
y = x
z = x + ix, x ∈ R
б) 2·|z| - 8z + 1 + 2i = 0
z = x + iy
2√(x² + y²) - 8·(x + iy) + 1 + 2i = 0
2√(x² + y²) - 8x - i8y + 1 + 2i = 0
2√(x² + y²) = (8x - 1) + i(8y - 2)
2√(x² + y²) = 8x - 1
8y - 2 = 0
y = 1/4
2√(x² + (1/4)²) = 8x - 1
4(x² + 1/16) = 64x² - 16x + 1
8x - 1 ≥ 1/2
4x² + 1/4 = 64x² - 16x + 1
8x ≥ 3/2
60x² - 16x + 3/4 = 0
x ≥ 3/16
240x² - 64x + 3 = 0
D = 64² - 4·240·3 = 1216
x = (64 (+/-) √1216)/480 = (64 (+/-) 8√19)/480 = (8 (+/-) √19)/60
x = 2/15 (+/-) √19/60
x ≥ 3/16
x = 2/15 + √19/60
z = 2/15 + √19/60 + i/4
{y=x
y²+y²=2
2y²=2
y²=1
y₁=1
y₂= -1
x₁=1
x₂= -1