Let, (a + ib)² = 1 + 4i

Now using, (a + b)² = a² + b² + 2ab

a² + (bi)² + 2abi = 1 + 4i

Since i² = -1

a² - b² + 2abi = 1 + 4i

Now, separating real and complex parts, we get

a² - b² = 1…………..eq.1

2ab =4…….eq.2

a =

Now, using the value of a in eq.1, we get

– b² = 1

12 – b⁴ = b²

b⁴ + b² - 12= 0

Simplify and get the value of b² , we get,

b² = -4 or b² = 3

as b is real no. so, b² = 3

b= or b=

Therefore, a= 2 or a= -2

Hence the square root of the complex no. is 2 + i and -2 -i.