### Video Transcript

The complex numbers π plus ππ
and π plus ππ, where π, π, π, and π are real numbers, are the roots of a
quadratic equation with real coefficients.

Given that π is not equal to zero,
what conditions, if any, must π, π, π, and π satisfy?

In this question, we are told that
π plus ππ and π plus ππ are roots to our quadratic equation with real
coefficients. This equation would usually be of
the form ππ₯ squared plus ππ₯ plus π equals zero, though π, π, and π are not
to be confused with the letters π, π, and π in our complex numbers. So weβll rewrite this as ππ₯
squared plus ππ₯ plus π equals zero.

Now we know that the nonreal roots
of a quadratic equation with real coefficients occur in complex conjugate pairs. And remember, to find the
conjugate, we change the sign of the imaginary part. So the conjugate of π plus ππ is
π minus ππ. And π plus ππ and π plus ππ
must be complex conjugates of one another by this theorem. This means that the conjugate of π
plus ππ must be equal to π plus ππ. So we say that π minus ππ equals
π plus ππ.

And for two complex numbers to be
equal, their real parts must be equal. So here we equate π and π. But their imaginary parts must also
be equal. So we equate the imaginary
parts. And we see that negative π equals
π. So the conditions that π, π, π,
and π must satisfy here is that π must be equal to π and negative π must be
equal to π. This time, weβre going to use our
knowledge of the nature of complex roots of quadratic equations to reconstruct an
equation given one of its roots.