Final answer:
The given equation z⁴ +1=0 can be factored using the difference of squares, which ultimately gives the roots z = i, z = -i, z = 1, and z = -1 using the imaginary unit i.
Step-by-step explanation:
The equation given is z⁴ +1=0. To solve this, we initially look for the roots of the equation. This is not a quadratic equation we can directly apply the quadratic formula to, but we can use the concept of complex numbers and factor the expression using the difference of squares.
The equation can be factored as:
(z² + 1)(z² - 1) = 0
This further simplifies to:
(z² + 1)(z + 1)(z - 1) = 0
However, we know that z² + 1 still can't be factored over the real numbers.
So we use the imaginary unit i, where i² = -1, to express the solutions for z² + 1 = 0, that are z=i and z=-i.
The solutions for z² - 1 = 0 are z=1 and z=-1.
Finally, the solutions for the equation z⁴ + 1 = 0 are: