Solve (-3+2i)/(3-6i)

$$( - 3 + 2 i ) / ( 3 - 6 i )$$

Evaluate

$$-\frac{7}{15}-\frac{4}{15}i\approx -0.466666667-0.266666667i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 3+6i.

$$\frac{\left(-3+2i\right)\left(3+6i\right)}{\left(3-6i\right)\left(3+6i\right)}$$

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

$$\frac{\left(-3+2i\right)\left(3+6i\right)}{3^{2}-6^{2}i^{2}}$$

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

$$\frac{\left(-3+2i\right)\left(3+6i\right)}{45}$$

Multiply complex numbers -3+2i and 3+6i like you multiply binomials.

$$\frac{-3\times 3-3\times \left(6i\right)+2i\times 3+2\times 6i^{2}}{45}$$

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

$$\frac{-3\times 3-3\times \left(6i\right)+2i\times 3+2\times 6\left(-1\right)}{45}$$

Do the multiplications in -3\times 3-3\times \left(6i\right)+2i\times 3+2\times 6\left(-1\right).

$$\frac{-9-18i+6i-12}{45}$$

Combine the real and imaginary parts in -9-18i+6i-12.

$$\frac{-9-12+\left(-18+6\right)i}{45}$$

Do the additions in -9-12+\left(-18+6\right)i.

$$\frac{-21-12i}{45}$$

Divide -21-12i by 45 to get -\frac{7}{15}-\frac{4}{15}i.

$$-\frac{7}{15}-\frac{4}{15}i$$

Real Part

$$-\frac{7}{15} = -0.4666666666666667$$

Solution Steps

Multiply both numerator and denominator of \frac{-3+2i}{3-6i} by the complex conjugate of the denominator, 3+6i.

$$Re(\frac{\left(-3+2i\right)\left(3+6i\right)}{\left(3-6i\right)\left(3+6i\right)})$$

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

$$Re(\frac{\left(-3+2i\right)\left(3+6i\right)}{3^{2}-6^{2}i^{2}})$$

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

$$Re(\frac{\left(-3+2i\right)\left(3+6i\right)}{45})$$

Multiply complex numbers -3+2i and 3+6i like you multiply binomials.

$$Re(\frac{-3\times 3-3\times \left(6i\right)+2i\times 3+2\times 6i^{2}}{45})$$

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

$$Re(\frac{-3\times 3-3\times \left(6i\right)+2i\times 3+2\times 6\left(-1\right)}{45})$$

Do the multiplications in -3\times 3-3\times \left(6i\right)+2i\times 3+2\times 6\left(-1\right).

$$Re(\frac{-9-18i+6i-12}{45})$$

Combine the real and imaginary parts in -9-18i+6i-12.

$$Re(\frac{-9-12+\left(-18+6\right)i}{45})$$

Do the additions in -9-12+\left(-18+6\right)i.

$$Re(\frac{-21-12i}{45})$$

Divide -21-12i by 45 to get -\frac{7}{15}-\frac{4}{15}i.

$$Re(-\frac{7}{15}-\frac{4}{15}i)$$

The real part of -\frac{7}{15}-\frac{4}{15}i is -\frac{7}{15}.

$$-\frac{7}{15}$$