The complex numbers 𝑎 plus 𝑏𝑖
and 𝑐 plus 𝑑𝑖, where 𝑎, 𝑏, 𝑐, and 𝑑 are real numbers, are the roots of a
quadratic equation with real coefficients.
Given that 𝑏 is not equal to zero,
what conditions, if any, must 𝑎, 𝑏, 𝑐, and 𝑑 satisfy?
In this question, we are told that
𝑎 plus 𝑏𝑖 and 𝑐 plus 𝑑𝑖 are roots to our quadratic equation with real
coefficients. This equation would usually be of
the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero, though 𝑎, 𝑏, and 𝑐 are not
to be confused with the letters 𝑎, 𝑏, and 𝑐 in our complex numbers. So we’ll rewrite this as 𝑝𝑥
squared plus 𝑞𝑥 plus 𝑟 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 𝑎 plus 𝑏𝑖 is
𝑎 minus 𝑏𝑖. And 𝑎 plus 𝑏𝑖 and 𝑐 plus 𝑑𝑖
must be complex conjugates of one another by this theorem. This means that the conjugate of 𝑎
plus 𝑏𝑖 must be equal to 𝑐 plus 𝑑𝑖. So we say that 𝑎 minus 𝑏𝑖 equals
𝑐 plus 𝑑𝑖.
And for two complex numbers to be
equal, their real parts must be equal. So here we equate 𝑎 and 𝑐. But their imaginary parts must also
be equal. So we equate the imaginary
parts. And we see that negative 𝑏 equals
𝑑. So the conditions that 𝑎, 𝑏, 𝑐,
and 𝑑 must satisfy here is that 𝑎 must be equal to 𝑐 and negative 𝑏 must be
equal to 𝑑. This time, we’re going to use our
knowledge of the nature of complex roots of quadratic equations to reconstruct an
equation given one of its roots.