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

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

Evaluate

$$-\frac{1}{25}-\frac{18}{25}i=-0.04-0.72i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 3+4i.

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

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

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

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

$$\frac{\left(-3-2i\right)\left(3+4i\right)}{25}$$

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

$$\frac{-3\times 3-3\times \left(4i\right)-2i\times 3-2\times 4i^{2}}{25}$$

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

$$\frac{-3\times 3-3\times \left(4i\right)-2i\times 3-2\times 4\left(-1\right)}{25}$$

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

$$\frac{-9-12i-6i+8}{25}$$

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

$$\frac{-9+8+\left(-12-6\right)i}{25}$$

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

$$\frac{-1-18i}{25}$$

Divide -1-18i by 25 to get -\frac{1}{25}-\frac{18}{25}i.

$$-\frac{1}{25}-\frac{18}{25}i$$

Real Part

$$-\frac{1}{25} = -0.04$$

Solution Steps

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

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

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

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

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

$$Re(\frac{\left(-3-2i\right)\left(3+4i\right)}{25})$$

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

$$Re(\frac{-3\times 3-3\times \left(4i\right)-2i\times 3-2\times 4i^{2}}{25})$$

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

$$Re(\frac{-3\times 3-3\times \left(4i\right)-2i\times 3-2\times 4\left(-1\right)}{25})$$

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

$$Re(\frac{-9-12i-6i+8}{25})$$

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

$$Re(\frac{-9+8+\left(-12-6\right)i}{25})$$

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

$$Re(\frac{-1-18i}{25})$$

Divide -1-18i by 25 to get -\frac{1}{25}-\frac{18}{25}i.

$$Re(-\frac{1}{25}-\frac{18}{25}i)$$

The real part of -\frac{1}{25}-\frac{18}{25}i is -\frac{1}{25}.

$$-\frac{1}{25}$$