Solve -4+20i/-6+4i

$$\frac { - 4 + 20 i } { - 6 + 4 i }$$

Evaluate

$$2-2i$$

Solution Steps

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

$$\frac{\left(-4+20i\right)\left(-6-4i\right)}{\left(-6+4i\right)\left(-6-4i\right)}$$

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

$$\frac{\left(-4+20i\right)\left(-6-4i\right)}{\left(-6\right)^{2}-4^{2}i^{2}}$$

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

$$\frac{\left(-4+20i\right)\left(-6-4i\right)}{52}$$

Multiply complex numbers -4+20i and -6-4i like you multiply binomials.

$$\frac{-4\left(-6\right)-4\times \left(-4i\right)+20i\left(-6\right)+20\left(-4\right)i^{2}}{52}$$

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

$$\frac{-4\left(-6\right)-4\times \left(-4i\right)+20i\left(-6\right)+20\left(-4\right)\left(-1\right)}{52}$$

Do the multiplications in -4\left(-6\right)-4\times \left(-4i\right)+20i\left(-6\right)+20\left(-4\right)\left(-1\right).

$$\frac{24+16i-120i+80}{52}$$

Combine the real and imaginary parts in 24+16i-120i+80.

$$\frac{24+80+\left(16-120\right)i}{52}$$

Do the additions in 24+80+\left(16-120\right)i.

$$\frac{104-104i}{52}$$

Divide 104-104i by 52 to get 2-2i.

$$2-2i$$

Real Part

$$2$$

Solution Steps

Multiply both numerator and denominator of \frac{-4+20i}{-6+4i} by the complex conjugate of the denominator, -6-4i.

$$Re(\frac{\left(-4+20i\right)\left(-6-4i\right)}{\left(-6+4i\right)\left(-6-4i\right)})$$

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

$$Re(\frac{\left(-4+20i\right)\left(-6-4i\right)}{\left(-6\right)^{2}-4^{2}i^{2}})$$

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

$$Re(\frac{\left(-4+20i\right)\left(-6-4i\right)}{52})$$

Multiply complex numbers -4+20i and -6-4i like you multiply binomials.

$$Re(\frac{-4\left(-6\right)-4\times \left(-4i\right)+20i\left(-6\right)+20\left(-4\right)i^{2}}{52})$$

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

$$Re(\frac{-4\left(-6\right)-4\times \left(-4i\right)+20i\left(-6\right)+20\left(-4\right)\left(-1\right)}{52})$$

Do the multiplications in -4\left(-6\right)-4\times \left(-4i\right)+20i\left(-6\right)+20\left(-4\right)\left(-1\right).

$$Re(\frac{24+16i-120i+80}{52})$$

Combine the real and imaginary parts in 24+16i-120i+80.

$$Re(\frac{24+80+\left(16-120\right)i}{52})$$

Do the additions in 24+80+\left(16-120\right)i.

$$Re(\frac{104-104i}{52})$$

Divide 104-104i by 52 to get 2-2i.

$$Re(2-2i)$$

The real part of 2-2i is 2.

$$2$$