Solve 2+4i/2-5i

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

Evaluate

$$-\frac{16}{29}+\frac{18}{29}i\approx -0.551724138+0.620689655i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 2+5i.

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

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

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

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

$$\frac{\left(2+4i\right)\left(2+5i\right)}{29}$$

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

$$\frac{2\times 2+2\times \left(5i\right)+4i\times 2+4\times 5i^{2}}{29}$$

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

$$\frac{2\times 2+2\times \left(5i\right)+4i\times 2+4\times 5\left(-1\right)}{29}$$

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

$$\frac{4+10i+8i-20}{29}$$

Combine the real and imaginary parts in 4+10i+8i-20.

$$\frac{4-20+\left(10+8\right)i}{29}$$

Do the additions in 4-20+\left(10+8\right)i.

$$\frac{-16+18i}{29}$$

Divide -16+18i by 29 to get -\frac{16}{29}+\frac{18}{29}i.

$$-\frac{16}{29}+\frac{18}{29}i$$

Real Part

$$-\frac{16}{29} = -0.5517241379310345$$

Solution Steps

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

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

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

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

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

$$Re(\frac{\left(2+4i\right)\left(2+5i\right)}{29})$$

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

$$Re(\frac{2\times 2+2\times \left(5i\right)+4i\times 2+4\times 5i^{2}}{29})$$

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

$$Re(\frac{2\times 2+2\times \left(5i\right)+4i\times 2+4\times 5\left(-1\right)}{29})$$

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

$$Re(\frac{4+10i+8i-20}{29})$$

Combine the real and imaginary parts in 4+10i+8i-20.

$$Re(\frac{4-20+\left(10+8\right)i}{29})$$

Do the additions in 4-20+\left(10+8\right)i.

$$Re(\frac{-16+18i}{29})$$

Divide -16+18i by 29 to get -\frac{16}{29}+\frac{18}{29}i.

$$Re(-\frac{16}{29}+\frac{18}{29}i)$$

The real part of -\frac{16}{29}+\frac{18}{29}i is -\frac{16}{29}.

$$-\frac{16}{29}$$