How does this work? And when do we need complex numbers?

You have learned that you cannot take the square root of a negative number, for example, √-2 doesn't exist in the real numbers, because the product of any two of the same real numbers multiplied together will always be positive.

3² = 3 × 3 = 9   and   (-3)² = (-3) × (-3) = 9 ∴ √9 = 3 or -3

So, how do we take the square root of a negative real number? √-9 = ?

We have defined a special number i as the square root of -1 or √-1.

So, to find √-9

Rewrite as √-1 × 9 = √-1 × √9 = i × 3 = 3i

In this case, the value of a, the real number part of the complex number, is 0 and the number is said to be purely imaginary.

0 + 3i = 3i

The quadratic equation x² + 0 = 0 would have 3i and its Interactive popup. Assistance may be required. complex conjugate as its solutions and can be solved by moving the constant to the other side of the equation, then taking the square roots of both sides:

Complex conjugate – the complex conjugate of a + bi is a - bi. This consists of changing the sign on the imaginary part of the complex number. The real number part is left unchanged.