Solve (4-3i)/(2+3i)

$$( 4 - 3 i ) / ( 2 + 3 i )$$

Evaluate

$$-\frac{1}{13}-\frac{18}{13}i\approx -0.076923077-1.384615385i$$

Solution Steps

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

$$\frac{\left(4-3i\right)\left(2-3i\right)}{\left(2+3i\right)\left(2-3i\right)}$$

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

$$\frac{\left(4-3i\right)\left(2-3i\right)}{2^{2}-3^{2}i^{2}}$$

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

$$\frac{\left(4-3i\right)\left(2-3i\right)}{13}$$

Multiply complex numbers 4-3i and 2-3i like you multiply binomials.

$$\frac{4\times 2+4\times \left(-3i\right)-3i\times 2-3\left(-3\right)i^{2}}{13}$$

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

$$\frac{4\times 2+4\times \left(-3i\right)-3i\times 2-3\left(-3\right)\left(-1\right)}{13}$$

Do the multiplications in 4\times 2+4\times \left(-3i\right)-3i\times 2-3\left(-3\right)\left(-1\right).

$$\frac{8-12i-6i-9}{13}$$

Combine the real and imaginary parts in 8-12i-6i-9.

$$\frac{8-9+\left(-12-6\right)i}{13}$$

Do the additions in 8-9+\left(-12-6\right)i.

$$\frac{-1-18i}{13}$$

Divide -1-18i by 13 to get -\frac{1}{13}-\frac{18}{13}i.

$$-\frac{1}{13}-\frac{18}{13}i$$

Real Part

$$-\frac{1}{13} = -0.07692307692307693$$

Solution Steps

Multiply both numerator and denominator of \frac{4-3i}{2+3i} by the complex conjugate of the denominator, 2-3i.

$$Re(\frac{\left(4-3i\right)\left(2-3i\right)}{\left(2+3i\right)\left(2-3i\right)})$$

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

$$Re(\frac{\left(4-3i\right)\left(2-3i\right)}{2^{2}-3^{2}i^{2}})$$

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

$$Re(\frac{\left(4-3i\right)\left(2-3i\right)}{13})$$

Multiply complex numbers 4-3i and 2-3i like you multiply binomials.

$$Re(\frac{4\times 2+4\times \left(-3i\right)-3i\times 2-3\left(-3\right)i^{2}}{13})$$

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

$$Re(\frac{4\times 2+4\times \left(-3i\right)-3i\times 2-3\left(-3\right)\left(-1\right)}{13})$$

Do the multiplications in 4\times 2+4\times \left(-3i\right)-3i\times 2-3\left(-3\right)\left(-1\right).

$$Re(\frac{8-12i-6i-9}{13})$$

Combine the real and imaginary parts in 8-12i-6i-9.

$$Re(\frac{8-9+\left(-12-6\right)i}{13})$$

Do the additions in 8-9+\left(-12-6\right)i.

$$Re(\frac{-1-18i}{13})$$

Divide -1-18i by 13 to get -\frac{1}{13}-\frac{18}{13}i.

$$Re(-\frac{1}{13}-\frac{18}{13}i)$$

The real part of -\frac{1}{13}-\frac{18}{13}i is -\frac{1}{13}.

$$-\frac{1}{13}$$