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Chapter 2: Problem 26

Divide and express the result in standard form. $$\frac{-6 i}{3+2 i}$$

Short Answer

Expert verified
The answer is \(-(18/5)i+12/5\).

Step by step solution

01

Find the conjugate of the denominator

The conjugate of \(3+2i\) is \(3-2i\). The conjugate of a complex number \(a+bi\) is obtained by changing the sign of its imaginary part, hence \(a-bi\).
02

Multiply the numerator and the denominator by the conjugate of the denominator

Performing this operation: \(-6i \cdot (3-2i) / (3+2i) \cdot (3-2i)\). In the numerator, distribute \(-6i\) to both \(3\) and \(-2i\). In the denominator, use the formula \((a-b)(a+b)=a^2 -b^2\). After calculation this gives us: \(-18i+12 / 9-4\)
03

Simplify the expression

Now, simplify the denominator and the numerator separately. The denominator gives us \(5\). In the numerator, express the result in standard form \(a+bi\). This will leave us with \(0-18i+12/5\). After simplifying this expression we get \(-(18/5)i+12/5\) in the standard form.

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Most popular questions from this chapter

Solve each inequality using a graphing utility. $$\frac{1}{x+1} \leq \frac{2}{x+4}$$

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