### Video Transcript

Solve π§π§ star plus π§ star minus
π§ equals four plus two π.

Now, letβs just see what weβve got
here. π§ and π§ star denote complex
numbers. π§ itself will be in the form π
plus ππ. π and π are real constants. π represents the real part of the
complex number whereas π represents its imaginary part. π§ star is the complex
conjugate. And thatβs sometimes represented
using π bar as well. Now, to find the complex conjugate
of a complex number, we simply change the sign of the imaginary parts. And so if π§, our complex number,
is π plus ππ, π§ star is π minus ππ.

And there are some interesting
properties of a complex number and its conjugate. Firstly, when we find their
product, we end up with a purely real number. So letβs replace π§ with π plus
ππ and π§ star with π minus ππ in our complex equation. When we do, we find π§π§ star to be
π plus ππ times π minus ππ. And then, we add π minus ππ and
subtract π§, which is π plus ππ. And of course, this is all equal to
four plus two π.

Letβs distribute some of our
parentheses. Weβll begin here by multiplying the
first term in each expression. Thatβs π times π, which is equal
to π squared. Weβll multiply the outer term in
each expression. That gives us negative πππ. Weβll then multiply the inner terms
to get πππ. And finally, weβll multiply the
last term. That gives us negative ππ all
squared or negative π squared π squared plus π minus ππ just remains the
same. And then, weβre subtracting that
third term, that third expression π plus ππ. So we get minus π minus ππ.

Letβs see if we can neaten this up
a little bit. We can see that π minus π is
zero. So those cancel. Similarly, negative πππ plus
πππ is zero. We then recall that π squared is
equal to negative one. So we use this. And we write negative π squared π
squared as negative negative one π squared. Similarly, we can collect like
terms. And we have negative two ππ
here. Simplifying a little further and we
find π squared plus π squared minus two ππ is all equal to four plus two π.

Now, this bit is really
important. We essentially have two complex
numbers. On the left-hand side, the real
part of the complex number is simply π squared plus π squared whereas its
imaginary part, remember, thatβs the coefficient of π, is negative two π. On the right-hand side, the real
part of our complex number is four. And its imaginary part is two. And so, for the complex numbers on
each side of our equation to be equal, their real parts must be equal. And their imaginary parts must
separately be equal. That is, π squared plus π squared
must be equal to four. And negative two π must be equal
to two.

Well, we can now solve that second
equation. Weβll divide through by negative
two. And when we do, we find that π is
equal to negative one. Letβs substitute that into our
first equation. When we do, we find π squared plus
negative one squared equals four. Well, negative one squared is
simply one.

Weβll subtract one from both sides
of this equation to find that π squared is equal to three. And then, we square root both
sides, remembering that weβre going to need to take both the positive and negative
square root of three. And we find π is either positive
or negative root three. Now, if we go back to our original
equation, we said that π§ was equal to π plus ππ. Well, π is either positive or
negative root three. And π is negative one.

So this means the two solutions to
our equation are π§ equals root three minus π or π§ equals negative root three
minus π.