# Write a quadratic equation with imaginary numbers definition

So this right over here is going to be 2i. This and this or this and this, or this. This is 3 plus or minus i over 2. All I did-- you can see that this is just dividing both of these by 2.

### Find quadratic equation given complex roots calculator

So it also checks out. Once again, a little hairy. That's if I take the positive version of the i there. The imaginary unit number is i. If x can be any complex number, then y x may be complex itself. These roots are identical except for the "sign" separating the two terms. So plus 6i. Let's now take a closer look at how these numbers relate to quadratics. Wait a moment - did you notice something wrong there?

Let's now take a closer look at how these numbers relate to quadratics. And if that doesn't make sense to you, I encourage you to kind of multiply it out either with the distributive property or FOIL it out, and you'll get the middle term.

Times 2 over here, plus 5, needs to be equal to-- well, before I even multiply it out, we could divide the numerator and the denominator by 2.

So 6 divided by 2 is 3.

And the quadratic formula tells us that if we have something in standard form like this, that the roots of it are going to be negative b plus or minus-- so that gives us two roots right over there-- plus or minus square root of b squared minus 4ac over 2a.

So let's do that. Or you could go directly from this. So what we want to do is we want to take 2 times this quantity squared. The discriminant, b2 - 4ac, offers valuable information about the "nature" of the roots of a quadratic equation where a, b and c are rational values.

### Complex quadratic equation

Once again, a little hairy. But just to put it into a form that we're more familiar with, let's try to put it into standard form. Let me rewrite the original equation. So negative b is negative negative 6. So negative i squared is also negative 1. These are equivalent. You'll have to either imagine this axis coming out of the paper, or draw it in perspective. This and this or this and this, or this. We're mainly interested in the roots of the quadratic - where the "surface" y x crosses the x-plane. Did you notice One way to picture it is to look at the real and imaginary parts of y x separately. All of that over 2 times a. This and these two guys right here are equivalent. Minus 1. Upon investigation, it was discovered that these square roots were called imaginary numbers and the roots were referred to as complex roots.

These are all equal representations of both of the roots. So it also checks out.

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