Video Transcript
Solve two plus three 𝑖 times 𝑧
squared plus four 𝑧 minus six 𝑖 plus four equals zero.
We use the quadratic formula. We substitute the values of the
coefficients for 𝑎, 𝑏, and 𝑐. And now we simplify. In the denominator, we get four
plus six 𝑖. In the numerator, we get negative
four plus or minus a big radical. And inside that radical, four
squared is 16. And from that, we subtract four
times the product of 𝑎 and 𝑐. When we distribute, we see that the
terms involving 𝑖 cancel. And we’re left with just eight plus
18, which is 26. 16 minus four times 26 is negative
88.
Notice that we have a negative
discriminant here. The square root of negative 88 is
𝑖 times the square root of 88 or two 𝑖 times the square root of 22. It’s tempting to say that this is
our final answer here. But we’d like our denominators to
be real if possible. We do that by multiplying the
numerator and denominator by the complex conjugate of this denominator.
Let’s do this for the first
root. We distribute in the numerator and
in the denominator too. And in the denominator, we notice
that the terms involving 𝑖 cancel, leaving just the modulus of our complex number
in the denominator. That’s four squared plus six
squared. We evaluate the denominator and
group the real and imaginary parts together in the numerator. And we notice that we can cancel a
factor of four. We can therefore write our first
root in simplest form as shown. And we can use exactly the same
procedure to simplify the second root.
Notice that although our quadratic
had a negative discriminant, these two roots are not complex conjugates. They don’t even have the same real
parts.
