Solve 73+2i/2-5i+3-2i/2+5i

$$7 \frac{ 3+2i }{ 2-5i } + \frac{ 3-2i }{ 2+5i }$$

Evaluate

$$-\frac{32}{29}+\frac{114}{29}i\approx -1.103448276+3.931034483i$$

Solution Steps

Multiply both numerator and denominator of \frac{3+2i}{2-5i} by the complex conjugate of the denominator, 2+5i.

$$7\times \frac{\left(3+2i\right)\left(2+5i\right)}{\left(2-5i\right)\left(2+5i\right)}+\frac{3-2i}{2+5i}$$

Do the multiplications in \frac{\left(3+2i\right)\left(2+5i\right)}{\left(2-5i\right)\left(2+5i\right)}.

$$7\times \frac{-4+19i}{29}+\frac{3-2i}{2+5i}$$

Divide -4+19i by 29 to get -\frac{4}{29}+\frac{19}{29}i.

$$7\left(-\frac{4}{29}+\frac{19}{29}i\right)+\frac{3-2i}{2+5i}$$

Multiply 7 and -\frac{4}{29}+\frac{19}{29}i to get -\frac{28}{29}+\frac{133}{29}i.

$$-\frac{28}{29}+\frac{133}{29}i+\frac{3-2i}{2+5i}$$

Multiply both numerator and denominator of \frac{3-2i}{2+5i} by the complex conjugate of the denominator, 2-5i.

$$-\frac{28}{29}+\frac{133}{29}i+\frac{\left(3-2i\right)\left(2-5i\right)}{\left(2+5i\right)\left(2-5i\right)}$$

Do the multiplications in \frac{\left(3-2i\right)\left(2-5i\right)}{\left(2+5i\right)\left(2-5i\right)}.

$$-\frac{28}{29}+\frac{133}{29}i+\frac{-4-19i}{29}$$

Divide -4-19i by 29 to get -\frac{4}{29}-\frac{19}{29}i.

$$-\frac{28}{29}+\frac{133}{29}i+\left(-\frac{4}{29}-\frac{19}{29}i\right)$$

Add -\frac{28}{29}+\frac{133}{29}i and -\frac{4}{29}-\frac{19}{29}i to get -\frac{32}{29}+\frac{114}{29}i.

$$-\frac{32}{29}+\frac{114}{29}i$$

Real Part

$$-\frac{32}{29} = -1\frac{3}{29} = -1.103448275862069$$

Solution Steps

Multiply both numerator and denominator of \frac{3+2i}{2-5i} by the complex conjugate of the denominator, 2+5i.

$$Re(7\times \frac{\left(3+2i\right)\left(2+5i\right)}{\left(2-5i\right)\left(2+5i\right)}+\frac{3-2i}{2+5i})$$

Do the multiplications in \frac{\left(3+2i\right)\left(2+5i\right)}{\left(2-5i\right)\left(2+5i\right)}.

$$Re(7\times \frac{-4+19i}{29}+\frac{3-2i}{2+5i})$$

Divide -4+19i by 29 to get -\frac{4}{29}+\frac{19}{29}i.

$$Re(7\left(-\frac{4}{29}+\frac{19}{29}i\right)+\frac{3-2i}{2+5i})$$

Multiply 7 and -\frac{4}{29}+\frac{19}{29}i to get -\frac{28}{29}+\frac{133}{29}i.

$$Re(-\frac{28}{29}+\frac{133}{29}i+\frac{3-2i}{2+5i})$$

Multiply both numerator and denominator of \frac{3-2i}{2+5i} by the complex conjugate of the denominator, 2-5i.

$$Re(-\frac{28}{29}+\frac{133}{29}i+\frac{\left(3-2i\right)\left(2-5i\right)}{\left(2+5i\right)\left(2-5i\right)})$$

Do the multiplications in \frac{\left(3-2i\right)\left(2-5i\right)}{\left(2+5i\right)\left(2-5i\right)}.

$$Re(-\frac{28}{29}+\frac{133}{29}i+\frac{-4-19i}{29})$$

Divide -4-19i by 29 to get -\frac{4}{29}-\frac{19}{29}i.

$$Re(-\frac{28}{29}+\frac{133}{29}i+\left(-\frac{4}{29}-\frac{19}{29}i\right))$$

Add -\frac{28}{29}+\frac{133}{29}i and -\frac{4}{29}-\frac{19}{29}i to get -\frac{32}{29}+\frac{114}{29}i.

$$Re(-\frac{32}{29}+\frac{114}{29}i)$$

The real part of -\frac{32}{29}+\frac{114}{29}i is -\frac{32}{29}.

$$-\frac{32}{29}$$