Solve -4-6i/6+4i

$$\frac { - 4 - 6 i } { 6 + 4 i }$$

Evaluate

$$-\frac{12}{13}-\frac{5}{13}i\approx -0.923076923-0.384615385i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 6-4i.

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

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

$$\frac{\left(-4-6i\right)\left(6-4i\right)}{6^{2}-4^{2}i^{2}}$$

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

$$\frac{\left(-4-6i\right)\left(6-4i\right)}{52}$$

Multiply complex numbers -4-6i and 6-4i like you multiply binomials.

$$\frac{-4\times 6-4\times \left(-4i\right)-6i\times 6-6\left(-4\right)i^{2}}{52}$$

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

$$\frac{-4\times 6-4\times \left(-4i\right)-6i\times 6-6\left(-4\right)\left(-1\right)}{52}$$

Do the multiplications in -4\times 6-4\times \left(-4i\right)-6i\times 6-6\left(-4\right)\left(-1\right).

$$\frac{-24+16i-36i-24}{52}$$

Combine the real and imaginary parts in -24+16i-36i-24.

$$\frac{-24-24+\left(16-36\right)i}{52}$$

Do the additions in -24-24+\left(16-36\right)i.

$$\frac{-48-20i}{52}$$

Divide -48-20i by 52 to get -\frac{12}{13}-\frac{5}{13}i.

$$-\frac{12}{13}-\frac{5}{13}i$$

Real Part

$$-\frac{12}{13} = -0.9230769230769231$$

Solution Steps

Multiply both numerator and denominator of \frac{-4-6i}{6+4i} by the complex conjugate of the denominator, 6-4i.

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

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

$$Re(\frac{\left(-4-6i\right)\left(6-4i\right)}{6^{2}-4^{2}i^{2}})$$

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

$$Re(\frac{\left(-4-6i\right)\left(6-4i\right)}{52})$$

Multiply complex numbers -4-6i and 6-4i like you multiply binomials.

$$Re(\frac{-4\times 6-4\times \left(-4i\right)-6i\times 6-6\left(-4\right)i^{2}}{52})$$

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

$$Re(\frac{-4\times 6-4\times \left(-4i\right)-6i\times 6-6\left(-4\right)\left(-1\right)}{52})$$

Do the multiplications in -4\times 6-4\times \left(-4i\right)-6i\times 6-6\left(-4\right)\left(-1\right).

$$Re(\frac{-24+16i-36i-24}{52})$$

Combine the real and imaginary parts in -24+16i-36i-24.

$$Re(\frac{-24-24+\left(16-36\right)i}{52})$$

Do the additions in -24-24+\left(16-36\right)i.

$$Re(\frac{-48-20i}{52})$$

Divide -48-20i by 52 to get -\frac{12}{13}-\frac{5}{13}i.

$$Re(-\frac{12}{13}-\frac{5}{13}i)$$

The real part of -\frac{12}{13}-\frac{5}{13}i is -\frac{12}{13}.

$$-\frac{12}{13}$$