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)

jernplate8 2021-09-16 Answered
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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Expert Answer

Tuthornt
Answered 2021-09-17 Author has 13696 answers
Step 1
Given:
\(\displaystyle{\frac{{{10}+\sqrt{{-{45}}}}}{{{4}}}}\)
Step 2
\(\displaystyle{\frac{{{10}+\sqrt{{-{45}}}}}{{{4}}}}={\frac{{{10}+{i}\sqrt{{{45}}}}}{{{4}}}}{\left(\sqrt{{-{1}}}={i}\right)}\)
\(\displaystyle={\frac{{{10}+{i}\sqrt{{{9}\times{5}}}}}{{{4}}}}\)
\(\displaystyle={\frac{{{10}+{i}{3}\sqrt{{{5}}}}}{{{4}}}}\)
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