Why do we need complex numbers, what are conjugates, and why do we need them

I guess I can give a brief overview and then you can decide if I should go deeper into these meaning and use.

Complex numbers are useful because in a lot of cases we need to use the square root of a negative number. Ex.

Solve the quadratic equation x^2+x+1=0

We have to use the quadratic formula since its not factorable.

x=[-1+-sqrt(1-4*1*1)]/2

x=-1/2 + or - sqrt(-3)

If we don't have imaginary numbers, we would be stuck because sqrt(-3) wouldn't exist as you cannot take the square root of a negative number.

But with complex numbers we can actually get the answer for that equation as

x=-1/2 + or - sqrt3*i

Which would give us a mean of using this answer to calculate further to get other needs depending on the problem you are solving. If complex number is not a thing, about half of our equations will have undefined answers.

Now conjugates is a more interesting one, on the surface level we are discussing here there is not much its used for except if you have a complex number as a root of a quadratic equation, then the conjugate of that complex number must also be a solution. Again if you want me to go a more in depth just comment down below, because I could write an entire essay on this topic :-)