# Question Video: Quadratic Equations with Complex Roots Mathematics

The complex numbers π + ππ and π + ππ, where π, π, π, and π are real numbers, are the roots of a quadratic equation with real coefficients. Given that π β  0, what conditions, if any, must π, π, π, and π satisfy?

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### Video Transcript

The complex numbers π plus ππ and π plus ππ, 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 ππ and π plus ππ are roots to our quadratic equation with real coefficients. This equation would usually be of the form ππ₯ squared plus ππ₯ 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 weβll rewrite this as ππ₯ squared plus ππ₯ 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 ππ is π minus ππ. And π plus ππ and π plus ππ must be complex conjugates of one another by this theorem. This means that the conjugate of π plus ππ must be equal to π plus ππ. So we say that π minus ππ equals π plus ππ.

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, weβre going to use our knowledge of the nature of complex roots of quadratic equations to reconstruct an equation given one of its roots.