Question

Find all complex solutions of the equation 2^2 − 6 + 49 = 0. Express complex numbers in the form + .

Answer 1

To find the complex solutions of the equation we use the quadratic formula:

$\begin{gathered} x=\frac{-(-6)\pm\sqrt[]{(-6)^2-4(2)(49)}}{2(2)} \\ =\frac{6\pm\sqrt[]{36-392}}{4} \\ =\frac{6\pm\sqrt[]{-356}}{4} \\ =\frac{6\pm i\sqrt[]{356}}{4} \\ =\frac{6\pm i\sqrt[]{4\cdot89}}{4} \\ =\frac{6\pm i2\sqrt[]{89}}{4} \\ =\frac{3\pm i\sqrt[]{89}}{2} \end{gathered}$

Therefore the solutions of the equation are:

$\begin{gathered} x=\frac{3}{2}+\frac{\sqrt[]{89}}{2}i \\ \text{ or } \\ x=\frac{3}{2}-\frac{\sqrt[]{89}}{2}i \end{gathered}$