Question

One of the roots of a quadratic equation is 5 + 3i. Determine what the other roots are, if any, and describe what the graph of the equation will look like. Explain your answer using the fundamental theorem of algebra.

Answer 1

The fundamental theorem of algebra states that a polynomial equation of nth degree has n roots (distinct or coincident).
So if one root is already known (5+3i), then there is exact one other root for a total of 2 for a quadratic equation (degree 2).

Furthermore, assuming the coefficients of the quadratic equation are real, then any complex root is accompanied by its complex conjugate, meaning that the sum of the two roots is a real number.

For example, the complex conjugate of 5+3i is 5-3i, because 5+3i + 5-3i = 10, a real number.

So the (only) other root is 5-3i, namely the complex conjugate of the given root.

The graph will be such that it will not touch or cross the x-axis, since the roots are complex.

Answer 2

Answer:

The equation is quadratic. Based on the fundamental theorem of algebra, we know it must have two roots. Because complex roots come in conjugate pairs, the other root must be 5 − 3i. Because the quadratic equation has two complex solutions, its graph will never cross the x-axis.