Solve 3-6i/2+2i

$$\frac { 3 - 6 i } { 2 + 2 i }$$

Evaluate

$$-\frac{3}{4}-\frac{9}{4}i=-0.75-2.25i$$

Solution Steps

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

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

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

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

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

$$\frac{\left(3-6i\right)\left(2-2i\right)}{8}$$

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

$$\frac{3\times 2+3\times \left(-2i\right)-6i\times 2-6\left(-2\right)i^{2}}{8}$$

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

$$\frac{3\times 2+3\times \left(-2i\right)-6i\times 2-6\left(-2\right)\left(-1\right)}{8}$$

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

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

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

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

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

$$\frac{-6-18i}{8}$$

Divide -6-18i by 8 to get -\frac{3}{4}-\frac{9}{4}i.

$$-\frac{3}{4}-\frac{9}{4}i$$

Real Part

$$-\frac{3}{4} = -0.75$$

Solution Steps

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

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

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

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

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

$$Re(\frac{\left(3-6i\right)\left(2-2i\right)}{8})$$

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

$$Re(\frac{3\times 2+3\times \left(-2i\right)-6i\times 2-6\left(-2\right)i^{2}}{8})$$

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

$$Re(\frac{3\times 2+3\times \left(-2i\right)-6i\times 2-6\left(-2\right)\left(-1\right)}{8})$$

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

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

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

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

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

$$Re(\frac{-6-18i}{8})$$

Divide -6-18i by 8 to get -\frac{3}{4}-\frac{9}{4}i.

$$Re(-\frac{3}{4}-\frac{9}{4}i)$$

The real part of -\frac{3}{4}-\frac{9}{4}i is -\frac{3}{4}.

$$-\frac{3}{4}$$