## Precalculus (6th Edition) Blitzer

The standard form of the expression $\frac{2+3i}{2+i}$ is $\frac{7}{5}+\frac{4}{5}i$.
Consider the expression,$\frac{2+3i}{2+i}$ Multiply by the complex conjugate of the denominator in the numerator and the denominator. $\frac{2+3i}{2+i}=\frac{\left( 2+3i \right)}{\left( 2+i \right)}\cdot \frac{\left( 2-i \right)}{\left( 2-i \right)}$ Use the FOIL method. \begin{align} & \frac{2+3i}{2+i}=\frac{\left( 2+3i \right)\left( 2-i \right)}{\left( 2+i \right)\left( 2-i \right)} \\ & =\frac{4-2i+6i-3{{i}^{2}}}{4-2i+2i-{{i}^{2}}} \\ & =\frac{4+4i-3{{i}^{2}}}{4-{{i}^{2}}} \end{align} Replace the value ${{i}^{2}}=-1$. \begin{align} & \frac{2+3i}{2+i}=\frac{4+4i-3\left( -1 \right)}{4-\left( -1 \right)} \\ & =\frac{4+4i+3}{4+1} \end{align} Make a group of real and imaginary terms. \begin{align} & \frac{2+3i}{2+i}=\frac{\left( 4+3 \right)+4i}{5} \\ & =\frac{7+4i}{5} \end{align} Express the complex number in the standard form. $\frac{2+3i}{2+i}=\frac{7}{5}+\frac{4}{5}i$ Therefore, the standard form of the expression $\frac{2+3i}{2+i}$ is $\frac{7}{5}+\frac{4}{5}i$.