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Find a complex number z satisfying the e...

Find a complex number `z` satisfying the equation `z+sqrt(2)|z+1|+i=0.`

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Givne equation is ` z + sqrt(2)|(z +1)| +i =0.`
` Let" "z = x + iy" "…(i)`
`rArr x +iy +sqrt(2)|x + iy + 1| +I = 0`
`rArr x + i (1 + y) +sqrt(2)[sqrt (x + 1)^(2) + Y^(2)] = 0 `
`rArr x + i (1 + y) +sqrt(2)sqrt ((x^(2) + 2x + 1+ y^(2))) = 0`
`rArr x + sqrt(2)sqrt (x^(2) + 2x + 1+ y^(2)) = 0 `
`rArr x^(2) = 2(x^(2) + 2x + 1 + y^(2)`
`rArr x^(2) + 4x + 2y6(2) + 2 = 0 " " ...(ii) 1 + y = 0 `
`rArr y = - 1`
For `y = - 1," " x^(2) + 4x + 2 + 2 = 0` [ using Eq. (ii)]
`rArr x^(2) + 4x + 4 = 0 rArr (x + 2)^(2) = 0`
`rArr x + 2 = 0 rArr x = -2`
`:. z = x + iy = - 2 - i`
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