Extracting the root of a complex number

In this publication, we will look at how you can take the root of a complex number, and also how this can help in solving quadratic equations whose discriminant is less than zero.

Extracting the root of a complex number

Square root

As we know, it is impossible to take the root of a negative real number. But when it comes to complex numbers, this action can be performed. Let’s figure it out.

Let’s say we have a number z = -9. For √-9 there are two roots:

z₁ = √-9 = -3i

z₁ = √-9 = 3i

Let us check the obtained results by solving the equation z² =-9, not forgetting that i² =-1:

(-3i)² = (-3)² ⋅ i² = 9 ⋅ (-1) = -9

(3i)² = 3² ⋅ i² = 9 ⋅ (-1) = -9

Thus, we have proved that -3i и 3i are roots √-9.

The root of a negative number is usually written like this:

√-1 = ±i

√-4 = ±2i

√-9 = ±3i

√-16 = ±4i etc.

Root to the power of n

Suppose we are given equations of the form z = ⁿ√w… It has n roots (z₀, of₁, of₂,…, z_n-1), which can be calculated using the formula below:

|w| is the module of a complex number w;

φ – his argument

k is a parameter that takes the values: k = {0, 1, 2,…, n-1}.

Quadratic equations with complex roots

Extracting the root of a negative number changes the usual idea of ​​uXNUMXbuXNUMXb. If the discriminant (D) is less than zero, then there cannot be real roots, but they can be represented as complex numbers.

Example

Let’s solve the equation x² – 8x + 20 = 0.

Solution

a = 1, b = -8, c = 20

D = b² – 4ac = 64 – 80 = -16

D < 0, but we can still take the root of the negative discriminant:

√D = √-16 = ±4i

Now we can calculate the roots:

x_1,2 = (-b ± √D)/2a = (8 ± 4i)/2 = 4 ± 2i.

Therefore, the equation x² – 8x + 20 = 0 has two complex conjugate roots:

x₁ = 4 + 2i

x₂ = 4 – 2i