Question

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

Answer 1

Given that: z + `sqrt(2) |(z + 1)| + i` = 0

Let z = x + yi

∴ `(x + yi) + sqrt(2)|(x + yi + 1)| + i` = 0

⇒ `x + (y + 1)i + sqrt(2)|(x + 1) + yi|` = 0

⇒ `x + (y + 1)i + sqrt(2) sqrt((x + 1)^2 + y^2)` = 0

⇒ `x + (y + 1)i + sqrt(2) sqrt(x^2 + 2x + 1 + y^2)` = 0 + 0i

⇒ `x + sqrt(2) sqrt(x^2 + 2x + 1 + y^2)` = 0, y + 1 = 0

⇒ x = `- sqrt(2) sqrt(x^2 + 2x + 1 + y^2)` and y = –1

⇒ x² = 2(x² + 2x + 1 + y²)

⇒ x² = 2x² + 4x + 2 + 2y²

⇒ x² + 4x + 2 + 2y² = 0

⇒ x² + 4x + 2 + 2(–1)² = 0  .....[∵y = –1]

⇒ x² + 4x + 4 = 0

⇒ (x + 2)² = 0

⇒ x + 2 = 0

⇒ x = –2

Hence, z = x + yi = –2 – i.