Question

If one root of a quadratic equation is 6+2i, determine the other root and the equation.

Answer 1

Complex Roots of the Quadratic Equation

When a quadratic equation has complex roots, they come in conjugate pairs, that is, if one of the roots is a + bi, the other must be a - bi.

* This only applies if the equation has real coefficients.

Thus, if one of the roots is 6 + 2i, the other root is 6 - 2i

To find the equation, we use this known statement: If p and q are the roots of a second-degree equation, then its equation is:

$x^2-(p+q)+pq=0$

We must add the roots and then multiply them as follows:

6 + 2i + 6 - 2i = 12

$\begin{gathered} \mleft(6+2i\mright)(6-2i)=6^2-(2i)^2 \\ (6+2i)(6-2i)=6^2-4i^2 \\ \text{Given that:} \\ i^2=-1\colon \\ \mleft(6+2i\mright)\mleft(6-2i\mright)=36+4 \\ \mleft(6+2i\mright)\mleft(6-2i\mright)=40 \end{gathered}$

Thus, the equation is:

$x^2-12x+40=0$