What is the answer to 1285/8-60i=10^6M^2 ?

The answer to 1285/8-60i=10^6M^2 is

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1285/8-60i=10^6M^2

\frac{1285}{8} - 60 i = 1 0^{6} M^{2}

M = \frac{257 + 1600000 \frac{66049}{160000 0 ^{2}} + \frac{9}{2500000000}}{800 5} - \frac{3}{25 5 257 + 1600000 \frac{66049}{160000 0 ^{2}} + \frac{9}{2500000000}} i, M = - \frac{257 + 1600000 \frac{66049}{160000 0 ^{2}} + \frac{9}{2500000000}}{800 5} + \frac{3}{25 5 257 + 1600000 \frac{66049}{160000 0 ^{2}} + \frac{9}{2500000000}} i

Solution steps

\frac{1285}{8} - 60 i = 1 0^{6} M^{2}

Substitute

M = x + y i

\frac{1285}{8} - 60 i = 1 0^{6} (x + y i)^{2}

Expand

1 0^{6} (x + y i)^{2} : (1000000 x^{2} - 1000000 y^{2}) + 2000000 i x y

\frac{1285}{8} - 60 i = (1000000 x^{2} - 1000000 y^{2}) + 2000000 i x y

Complex numbers can be equal only if their real and imaginary parts are equalRewrite as system of equations:

[\frac{1285}{8} = 1000000 x^{2} - 1000000 y^{2} - 60 = 2000000 x y]

[\frac{1285}{8} = 1000000 x^{2} - 1000000 y^{2} - 60 = 2000000 x y] : x = \frac{257 + 1600000 \frac{66049}{160000 0 ^{2}} + \frac{9}{2500000000}}{800 5}, x = - \frac{257 + 1600000 \frac{66049}{160000 0 ^{2}} + \frac{9}{2500000000}}{800 5}, y = - \frac{3}{25 5 257 + 1600000 \frac{66049}{160000 0 ^{2}} + \frac{9}{2500000000}} y = \frac{3}{25 5 257 + 1600000 \frac{66049}{160000 0 ^{2}} + \frac{9}{2500000000}}

Substitute back

M = x + y i

M = \frac{257 + 1600000 \frac{66049}{160000 0 ^{2}} + \frac{9}{2500000000}}{800 5} - \frac{3}{25 5 257 + 1600000 \frac{66049}{160000 0 ^{2}} + \frac{9}{2500000000}} i, M = - \frac{257 + 1600000 \frac{66049}{160000 0 ^{2}} + \frac{9}{2500000000}}{800 5} + \frac{3}{25 5 257 + 1600000 \frac{66049}{160000 0 ^{2}} + \frac{9}{2500000000}} i

x + y i = (1 - i)^{2}

\frac{1 + z}{1 - z} = 8 i - 5

\frac{1 + 2 z}{3 + 2 z} = i

\frac{- 1}{b + i} = \frac{1}{- b + i}

(x + y) + (x - y) \cdot i = 2 + 5 \cdot i