Define the imaginary number 𝑖 as a root (solution) of the equation 𝑥2 = −1.

Given that ''(i')' is defined as a solution to the equation ''(x^2 = -1)'', evaluate the following expressions. (a) Calculate ''(i^2)''. [1] (b) Simplify ''(i^3 + i^4)''. [1] (c) Determine the value of ''(i^5 + i^6 + i^7 + i^8)''. [1]

Which of the following correctly defines the imaginary unit \(i\) and identifies the equation for which it is a solution?

A student states that the equation \(x^2 + 6x + 13 = 0\) has no real solutions. (a) Calculate the discriminant and verify that it is negative. (1 mark) (b) Complete the square to express the equation in the form \((x + a)^2 + b = 0\) where \(a\) and \(b\) are integers. (1 mark) (c) Rearrange this completed-square form to show that \(x = -3 \pm 2i\), where \(i\) is the imaginary unit defined by \(i^2 = -1\). (2 marks)