Question

Which describes the number and type of roots of the equation x^4 - 64 = 0

a. 2 real roots, 2 imaginary roots
b. 4 real roots
c. 3 real roots, 1 imaginary root
d. 4 imaginary roots

Answer 1

a. 2 real roots, 2 imaginary roots

Step-by-step explanation

The given equation is

`{x}^(4) - 64 = 0`

We rewrite as difference of two squares,

`( {x}^(2) )^(2) - {8}^(2) = 0`

We factor using difference of two squares to get;

`( {x}^(2) - 8)( {x}^(2) + 8) = 0`

We now use the zero product property to get:

`{x}^(2) = 8 \: or \: {x}^(2) = - 8`

Take the square root of both sides to get;

`{x} = \pm √(8) \: or \: {x}^(2) = \pm √( - 8)`

`{x} = \pm 2√(2) \: or \: {x} = \pm 2√( 2) i`

`{x} = - 2√(2) \: or \: {x} = 2√( 2)`

are two real roots.

`{x} = - 2√(2)i \: or \: {x} = 2√( 2) i`

are two imaginary roots.

The correct answer is A.