Solve 2-5i/4+6i

$$\frac { 2 - 5 i } { 4 + 6 i }$$

Evaluate

$$-\frac{11}{26}-\frac{8}{13}i\approx -0.423076923-0.615384615i$$

Solution Steps

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

$$\frac{\left(2-5i\right)\left(4-6i\right)}{\left(4+6i\right)\left(4-6i\right)}$$

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

$$\frac{\left(2-5i\right)\left(4-6i\right)}{4^{2}-6^{2}i^{2}}$$

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

$$\frac{\left(2-5i\right)\left(4-6i\right)}{52}$$

Multiply complex numbers 2-5i and 4-6i like you multiply binomials.

$$\frac{2\times 4+2\times \left(-6i\right)-5i\times 4-5\left(-6\right)i^{2}}{52}$$

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

$$\frac{2\times 4+2\times \left(-6i\right)-5i\times 4-5\left(-6\right)\left(-1\right)}{52}$$

Do the multiplications in 2\times 4+2\times \left(-6i\right)-5i\times 4-5\left(-6\right)\left(-1\right).

$$\frac{8-12i-20i-30}{52}$$

Combine the real and imaginary parts in 8-12i-20i-30.

$$\frac{8-30+\left(-12-20\right)i}{52}$$

Do the additions in 8-30+\left(-12-20\right)i.

$$\frac{-22-32i}{52}$$

Divide -22-32i by 52 to get -\frac{11}{26}-\frac{8}{13}i.

$$-\frac{11}{26}-\frac{8}{13}i$$

Real Part

$$-\frac{11}{26} = -0.4230769230769231$$

Solution Steps

Multiply both numerator and denominator of \frac{2-5i}{4+6i} by the complex conjugate of the denominator, 4-6i.

$$Re(\frac{\left(2-5i\right)\left(4-6i\right)}{\left(4+6i\right)\left(4-6i\right)})$$

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

$$Re(\frac{\left(2-5i\right)\left(4-6i\right)}{4^{2}-6^{2}i^{2}})$$

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

$$Re(\frac{\left(2-5i\right)\left(4-6i\right)}{52})$$

Multiply complex numbers 2-5i and 4-6i like you multiply binomials.

$$Re(\frac{2\times 4+2\times \left(-6i\right)-5i\times 4-5\left(-6\right)i^{2}}{52})$$

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

$$Re(\frac{2\times 4+2\times \left(-6i\right)-5i\times 4-5\left(-6\right)\left(-1\right)}{52})$$

Do the multiplications in 2\times 4+2\times \left(-6i\right)-5i\times 4-5\left(-6\right)\left(-1\right).

$$Re(\frac{8-12i-20i-30}{52})$$

Combine the real and imaginary parts in 8-12i-20i-30.

$$Re(\frac{8-30+\left(-12-20\right)i}{52})$$

Do the additions in 8-30+\left(-12-20\right)i.

$$Re(\frac{-22-32i}{52})$$

Divide -22-32i by 52 to get -\frac{11}{26}-\frac{8}{13}i.

$$Re(-\frac{11}{26}-\frac{8}{13}i)$$

The real part of -\frac{11}{26}-\frac{8}{13}i is -\frac{11}{26}.

$$-\frac{11}{26}$$