Solve 𝑧 squared minus four plus four 𝑖 𝑧 plus eight 𝑖 equals zero.
Here we have a quadratic equation with nonreal coefficients. To solve an equation of this form, we can apply the usual methods. We can use a quadratic formula. For a quadratic equation of the form 𝑎𝑧 squared plus 𝑏𝑧 plus 𝑐 equals zero, the solutions are given by negative 𝑏 plus or minus the square root of 𝑏 squared minus four 𝑎𝑐 all over two 𝑎. In this case, 𝑎 is the coefficient of 𝑧 squared. So it’s one. 𝑏 is the coefficient of 𝑧. So that’s negative four plus four 𝑖. And 𝑐 is the constant term. So here that’s eight 𝑖.
Let’s substitute all of these into our quadratic formula. When we do, we find that 𝑧 is equal to negative negative four plus four 𝑖 plus or minus the square root of negative four plus four 𝑖 all squared minus four times one times eight 𝑖 all over two times one.
Well, of course, it follows that negative negative four plus four 𝑖 is just four plus four 𝑖. We distribute the parentheses, remembering that a negative times a negative is a positive. So we end up actually multiplying four plus four 𝑖 by four plus four 𝑖 to get 16 plus 32𝑖 plus 16𝑖 squared. Then we have negative 32𝑖. That’s negative four times one times eight 𝑖. And of course, that’s all over two.
Now we can spot quite quickly that 32𝑖 minus 32𝑖 is zero. But we also know that 𝑖 squared is equal to negative one. And so
16𝑖 [16𝑖 squared] is the same as negative 16. And then we see that 16 minus 16 is zero. The discriminant of our equation is equal to zero. And that makes our life a lot easier. There’s simply one solution to our equation. It’s four plus four 𝑖 all over two, which simplifies to two plus two 𝑖. And so the solution to our equation is 𝑧 equals two plus two 𝑖.