Solve 5i/5+i+9/5-i

$$\frac { 5 i } { 5 + i } + \frac { 9 } { 5 - i }$$

Evaluate

$$\frac{25}{13}+\frac{17}{13}i\approx 1.923076923+1.307692308i$$

Solution Steps

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

$$\frac{5i\left(5-i\right)}{\left(5+i\right)\left(5-i\right)}+\frac{9}{5-i}$$

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

$$\frac{5+25i}{26}+\frac{9}{5-i}$$

Divide 5+25i by 26 to get \frac{5}{26}+\frac{25}{26}i.

$$\frac{5}{26}+\frac{25}{26}i+\frac{9}{5-i}$$

Multiply both numerator and denominator of \frac{9}{5-i} by the complex conjugate of the denominator, 5+i.

$$\frac{5}{26}+\frac{25}{26}i+\frac{9\left(5+i\right)}{\left(5-i\right)\left(5+i\right)}$$

Do the multiplications in \frac{9\left(5+i\right)}{\left(5-i\right)\left(5+i\right)}.

$$\frac{5}{26}+\frac{25}{26}i+\frac{45+9i}{26}$$

Divide 45+9i by 26 to get \frac{45}{26}+\frac{9}{26}i.

$$\frac{5}{26}+\frac{25}{26}i+\left(\frac{45}{26}+\frac{9}{26}i\right)$$

Add \frac{5}{26}+\frac{25}{26}i and \frac{45}{26}+\frac{9}{26}i to get \frac{25}{13}+\frac{17}{13}i.

$$\frac{25}{13}+\frac{17}{13}i$$

Real Part

$$\frac{25}{13} = 1\frac{12}{13} = 1.9230769230769231$$

Solution Steps

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

$$Re(\frac{5i\left(5-i\right)}{\left(5+i\right)\left(5-i\right)}+\frac{9}{5-i})$$

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

$$Re(\frac{5+25i}{26}+\frac{9}{5-i})$$

Divide 5+25i by 26 to get \frac{5}{26}+\frac{25}{26}i.

$$Re(\frac{5}{26}+\frac{25}{26}i+\frac{9}{5-i})$$

Multiply both numerator and denominator of \frac{9}{5-i} by the complex conjugate of the denominator, 5+i.

$$Re(\frac{5}{26}+\frac{25}{26}i+\frac{9\left(5+i\right)}{\left(5-i\right)\left(5+i\right)})$$

Do the multiplications in \frac{9\left(5+i\right)}{\left(5-i\right)\left(5+i\right)}.

$$Re(\frac{5}{26}+\frac{25}{26}i+\frac{45+9i}{26})$$

Divide 45+9i by 26 to get \frac{45}{26}+\frac{9}{26}i.

$$Re(\frac{5}{26}+\frac{25}{26}i+\left(\frac{45}{26}+\frac{9}{26}i\right))$$

Add \frac{5}{26}+\frac{25}{26}i and \frac{45}{26}+\frac{9}{26}i to get \frac{25}{13}+\frac{17}{13}i.

$$Re(\frac{25}{13}+\frac{17}{13}i)$$

The real part of \frac{25}{13}+\frac{17}{13}i is \frac{25}{13}.

$$\frac{25}{13}$$