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

$$( 2 + 4 i ) \div ( 2 - 3 i )$$

Evaluate

$$-\frac{8}{13}+\frac{14}{13}i\approx -0.615384615+1.076923077i$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

$$\frac{4+6i+8i-12}{13}$$

Combine the real and imaginary parts in 4+6i+8i-12.

$$\frac{4-12+\left(6+8\right)i}{13}$$

Do the additions in 4-12+\left(6+8\right)i.

$$\frac{-8+14i}{13}$$

Divide -8+14i by 13 to get -\frac{8}{13}+\frac{14}{13}i.

$$-\frac{8}{13}+\frac{14}{13}i$$

Real Part

$$-\frac{8}{13} = -0.6153846153846154$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

$$Re(\frac{4+6i+8i-12}{13})$$

Combine the real and imaginary parts in 4+6i+8i-12.

$$Re(\frac{4-12+\left(6+8\right)i}{13})$$

Do the additions in 4-12+\left(6+8\right)i.

$$Re(\frac{-8+14i}{13})$$

Divide -8+14i by 13 to get -\frac{8}{13}+\frac{14}{13}i.

$$Re(-\frac{8}{13}+\frac{14}{13}i)$$

The real part of -\frac{8}{13}+\frac{14}{13}i is -\frac{8}{13}.

$$-\frac{8}{13}$$