Video: Quadratic Equations with Complex Roots

The complex numbers π‘Ž + 𝑏𝑖 and 𝑐 + 𝑑𝑖, where π‘Ž, 𝑏, 𝑐, and 𝑑 are real numbers, are the roots of a quadratic equation with real coefficients. Given that 𝑏 β‰  0, what conditions, if any, must π‘Ž, 𝑏, 𝑐, and 𝑑 satisfy?

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Video Transcript

The complex numbers π‘Ž plus 𝑏𝑖 and 𝑐 plus 𝑑𝑖, where π‘Ž, 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 π‘Ž, 𝑏, 𝑐, and 𝑑 satisfy?

In this question, we are told that π‘Ž plus 𝑏𝑖 and 𝑐 plus 𝑑𝑖 are roots to our quadratic equation with real coefficients. This equation would usually be of the form π‘Žπ‘₯ squared plus 𝑏π‘₯ plus 𝑐 equals zero, though π‘Ž, 𝑏, and 𝑐 are not to be confused with the letters π‘Ž, 𝑏, and 𝑐 in our complex numbers. So we’ll rewrite this as 𝑝π‘₯ squared plus π‘žπ‘₯ plus π‘Ÿ 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‘Ž plus 𝑏𝑖 is π‘Ž minus 𝑏𝑖. And π‘Ž plus 𝑏𝑖 and 𝑐 plus 𝑑𝑖 must be complex conjugates of one another by this theorem. This means that the conjugate of π‘Ž plus 𝑏𝑖 must be equal to 𝑐 plus 𝑑𝑖. So we say that π‘Ž minus 𝑏𝑖 equals 𝑐 plus 𝑑𝑖.

And for two complex numbers to be equal, their real parts must be equal. So here we equate π‘Ž 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 π‘Ž, 𝑏, 𝑐, and 𝑑 must satisfy here is that π‘Ž 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.

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