Determine the solution set of a quarter 𝑦 squared plus 36 is equal to zero over the set of complex numbers.
We’ll solve this equation the same way we’d solve any equation. And then we’ll get on to the second part we’re asked about complex numbers as we get a bit further through.
So the first stage in our equation is to actually subtract 36 from each side. So this will give us a quarter 𝑦 squared is equal to negative 36. And then we’re actually gonna multiply each side of our equation by four. So we get 𝑦 squared is equal to negative 144. And then as we’re trying to solve for 𝑦, what we actually do is we take the square root of both sides, which leaves us with 𝑦 is equal to the square root of negative 144.
And now it’s at this point we actually consider the second part of the question, where it talks about being over the set of complex numbers. And this is because the square root of negative 144 would usually mean that we’d have to stop because we wouldn’t have any solutions. However, it is actually at this point we can introduce 𝑖 which is our imaginary number because this is equal to the square root of negative one.
And from this, we can actually form a relationship that’s gonna help us to actually solve our problem further. And that relationship is that if 𝑛 is a positive integer, then the square root of negative 𝑛 is gonna be equal to 𝑖 root 𝑛. So therefore, if we apply it to our equation, we’re gonna get 𝑦 is equal to 𝑖 root 144. And that’s because 144 is our 𝑛.
Okay, great, and now we can actually continue and solve the problem because we’re now going to get 𝑦 is equal to plus or minus 12𝑖. And we get that because the square root of 144 is 12. So therefore, we can say that the solution set is that 𝑦 is equal to negative 12𝑖 and 12𝑖.