# Some of the solution sets for quadratic equations in the next sections in this chapter will contain complex numbers such as frac(-4+sqrt(-12))(2)

Some of the solution sets for quadratic equations in the next sections in this chapter will contain complex numbers such as $$\displaystyle{\frac{{-{4}+\sqrt{{-{12}}}}}{{{2}}}}\ \text{ and }\ {\frac{{-{4}-\sqrt{{-{12}}}}}{{{2}}}}$$. We can simplify the first number as follows. $$\displaystyle{\frac{{-{4}+\sqrt{{-{12}}}}}{{{2}}}}$$
$$\displaystyle={\frac{{-{4}+{i}\sqrt{{{12}}}}}{{{2}}}}$$
$$\displaystyle={\frac{{-{4}+{2}{i}\sqrt{{3}}}}{{{2}}}}$$
$$\displaystyle={\frac{{{2}{\left(-{2}+{i}\sqrt{{3}}\right)}}}{{{2}}}}$$
$$\displaystyle=-{2}+{i}\sqrt{{3}}$$ Simplify each of the following complex numbers.
$$\displaystyle{\frac{{{10}+\sqrt{{-{45}}}}}{{{4}}}}$$

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Tuthornt
Step 1
Given:
$$\displaystyle{\frac{{{10}+\sqrt{{-{45}}}}}{{{4}}}}$$
Step 2
$$\displaystyle{\frac{{{10}+\sqrt{{-{45}}}}}{{{4}}}}={\frac{{{10}+{i}\sqrt{{{45}}}}}{{{4}}}}{\left(\sqrt{{-{1}}}={i}\right)}$$
$$\displaystyle={\frac{{{10}+{i}\sqrt{{{9}\times{5}}}}}{{{4}}}}$$
$$\displaystyle={\frac{{{10}+{i}{3}\sqrt{{{5}}}}}{{{4}}}}$$