Question

Find the square roots of – 6 + 8i

Answer 1

Let `sqrt(- 6 + 8"i")` = a + ib

Squaring on both sides

– 6 + 8i = (a + ib)²

– 6 + 8i = a² – b² + 2iab

Equating real and imaginary parts

a² – b² = – 6 and 2ab = 8

Now (a² + b²)² = (a² – b²)² + 4a²b²

= (– 6)² + (8)²

= 36 + 64 = 100

∴ a + b² = 10

Solving a² – b² = – 6 and a² + b² = 10

We get 2a² = 4, b² = 8

a² = 2

b² = `+-  2sqrt(2)`

a = `+-  sqrt(2)`

∴ `sqrt(- 6 + 8"i") = +-  sqrt(2) +- "i"2sqrt(2)`

= `+-  (sqrt(2) + "i" 2sqrt(2))`

Aliter:

Square root of – 6 + 8i

Let a + ib = – 6 + 8i

a = – 6, b = 8

|z| = `sqrt(6^2 + 8^2)`

= `sqrt(100)`

= 10

`sqrt("a" + "b") = +-  [sqrt((sqrt("a"^2 + "b"^2) + "a")/2) + "i" "b"/|"b"| sqrt((sqrt("a"^2 + "b"^2) - "a")/2)]`

= `+-  [sqrt((10 - 6)/2) + "i" sqrt((10 + 6)/2)]`

= `+- [sqrt(2) + "i" 2sqrt(2)]`