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.
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.
I would need to know the length of a pair of corresponding sides, so I could set up a proportion to solve for the unknown length. I would set the ratio of the known corresponding sides equal to the unknown measure, making sure I put corresponding measures from the triangles in the same positions in the ratios.
