Video: Solving Quadratic Equations with Imaginary Roots

Solve the equation 2𝑥² + 8 = 0 over the set of complex numbers.

02:47

Video Transcript

Solve the equation two 𝑥 squared plus eight equals zero over the set of complex numbers.

Okay, for this problem, it asks us to solve the equation. So we’re gonna solve the equation in the same way that we’d solve any equation. And then, we’ll have a look at the second part of the question a little bit later on. Okay, so let’s get on and solve the equation.

The first thing we’re gonna do is actually subtract eight from each side. And this gives us two 𝑥 squared is equal to negative eight. Okay now, let’s do the next step. So now, if we divide each side of the equation by two, we’re gonna get 𝑥 squared is equal to negative four. Okay great! And then, for the next stage, what we’d do is we’d actually take the square root of both sides. And what that will leave us with is 𝑥 is equal to plus or minus root negative four. Okay, so it’s at this point, we’d usually get it right. Okay, we can’t go any further with our equation.

However, our question actually says in the second part that it wants the equation solved over the set of complex numbers. Great, so what it actually means is we can actually now introduce imaginary number to actually help us solve this equation. But before we actually introduce our imaginary number, what we can actually do is we can apply a surd rule to actually help us a little bit further. And the surd rule that we’re gonna apply is that root 𝑎𝑏 is equal to root 𝑎 multiplied by root 𝑏. So what this is gonna give us when we apply our surd rule is that 𝑥 is equal to plus or minus root four multiplied by root negative one.

And the reason we have so chosen the root four root negative one is whenever we’re trying to actually break a surd down into its two constituent parts, we always choose the highest square number that goes into it. So in this case, that would be root four would be the highest square factor. Okay great! We’ve now got down to an answer. That’s 𝑥 is equal to plus or minus two root negative one.

It’s at this point we can now introduce our measuring number. And we can do that because 𝑖, our imaginary number, is equal to the root of negative one. So now that we know that 𝑖 is equal to the square root of negative one, we can actually substitute that in, which will give us 𝑥 is equal to plus or minus two 𝑖.

Therefore, we can say that the equation two 𝑥 squared plus eight equals zero solved over the set of complex numbers will give us that 𝑥 is equal to two 𝑖 and negative two 𝑖, where 𝑖 is an imaginary number equal to root negative one.

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