Solve (-17i-i):(2+3i)

$$( - 17 i - i ) : ( 2 + 3 i )$$

Evaluate

$$-\frac{54}{13}-\frac{36}{13}i\approx -4.153846154-2.769230769i$$

Solution Steps

Subtract i from -17i to get -18i.

$$\frac{-18i}{2+3i}$$

Multiply both numerator and denominator by the complex conjugate of the denominator, 2-3i.

$$\frac{-18i\left(2-3i\right)}{\left(2+3i\right)\left(2-3i\right)}$$

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{-18i\left(2-3i\right)}{2^{2}-3^{2}i^{2}}$$

By definition, i^{2} is -1. Calculate the denominator.

$$\frac{-18i\left(2-3i\right)}{13}$$

Multiply -18i times 2-3i.

$$\frac{-18i\times 2-18\left(-3\right)i^{2}}{13}$$

By definition, i^{2} is -1.

$$\frac{-18i\times 2-18\left(-3\right)\left(-1\right)}{13}$$

Do the multiplications in -18i\times 2-18\left(-3\right)\left(-1\right). Reorder the terms.

$$\frac{-54-36i}{13}$$

Divide -54-36i by 13 to get -\frac{54}{13}-\frac{36}{13}i.

$$-\frac{54}{13}-\frac{36}{13}i$$

Real Part

$$-\frac{54}{13} = -4\frac{2}{13} = -4.153846153846154$$

Solution Steps

Subtract i from -17i to get -18i.

$$Re(\frac{-18i}{2+3i})$$

Multiply both numerator and denominator of \frac{-18i}{2+3i} by the complex conjugate of the denominator, 2-3i.

$$Re(\frac{-18i\left(2-3i\right)}{\left(2+3i\right)\left(2-3i\right)})$$

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$Re(\frac{-18i\left(2-3i\right)}{2^{2}-3^{2}i^{2}})$$

By definition, i^{2} is -1. Calculate the denominator.

$$Re(\frac{-18i\left(2-3i\right)}{13})$$

Multiply -18i times 2-3i.

$$Re(\frac{-18i\times 2-18\left(-3\right)i^{2}}{13})$$

By definition, i^{2} is -1.

$$Re(\frac{-18i\times 2-18\left(-3\right)\left(-1\right)}{13})$$

Do the multiplications in -18i\times 2-18\left(-3\right)\left(-1\right). Reorder the terms.

$$Re(\frac{-54-36i}{13})$$

Divide -54-36i by 13 to get -\frac{54}{13}-\frac{36}{13}i.

$$Re(-\frac{54}{13}-\frac{36}{13}i)$$

The real part of -\frac{54}{13}-\frac{36}{13}i is -\frac{54}{13}.

$$-\frac{54}{13}$$