Solve 3{left(4+texttt{i}right)}^2-frac{-3+6texttt{i}}{4+texttt{i}}

$$3 { \left(4+ \texttt{i} \right) }^{ 2 } - \frac{ -3+6 \texttt{i} }{ 4+ \texttt{i} }$$

Evaluate

$$\frac{771}{17}+\frac{381}{17}i\approx 45.352941176+22.411764706i$$

Solution Steps

Calculate 4+i to the power of 2 and get 15+8i.

$$3\left(15+8i\right)-\frac{-3+6i}{4+i}$$

Multiply 3 and 15+8i to get 45+24i.

$$45+24i-\frac{-3+6i}{4+i}$$

Multiply both numerator and denominator of \frac{-3+6i}{4+i} by the complex conjugate of the denominator, 4-i.

$$45+24i-\frac{\left(-3+6i\right)\left(4-i\right)}{\left(4+i\right)\left(4-i\right)}$$

Do the multiplications in \frac{\left(-3+6i\right)\left(4-i\right)}{\left(4+i\right)\left(4-i\right)}.

$$45+24i-\frac{-6+27i}{17}$$

Divide -6+27i by 17 to get -\frac{6}{17}+\frac{27}{17}i.

$$45+24i+\left(\frac{6}{17}-\frac{27}{17}i\right)$$

Add 45+24i and \frac{6}{17}-\frac{27}{17}i to get \frac{771}{17}+\frac{381}{17}i.

$$\frac{771}{17}+\frac{381}{17}i$$

Real Part

$$\frac{771}{17} = 45\frac{6}{17} = 45.35294117647059$$

Solution Steps

Calculate 4+i to the power of 2 and get 15+8i.

$$Re(3\left(15+8i\right)-\frac{-3+6i}{4+i})$$

Multiply 3 and 15+8i to get 45+24i.

$$Re(45+24i-\frac{-3+6i}{4+i})$$

Multiply both numerator and denominator of \frac{-3+6i}{4+i} by the complex conjugate of the denominator, 4-i.

$$Re(45+24i-\frac{\left(-3+6i\right)\left(4-i\right)}{\left(4+i\right)\left(4-i\right)})$$

Do the multiplications in \frac{\left(-3+6i\right)\left(4-i\right)}{\left(4+i\right)\left(4-i\right)}.

$$Re(45+24i-\frac{-6+27i}{17})$$

Divide -6+27i by 17 to get -\frac{6}{17}+\frac{27}{17}i.

$$Re(45+24i+\left(\frac{6}{17}-\frac{27}{17}i\right))$$

Add 45+24i and \frac{6}{17}-\frac{27}{17}i to get \frac{771}{17}+\frac{381}{17}i.

$$Re(\frac{771}{17}+\frac{381}{17}i)$$

The real part of \frac{771}{17}+\frac{381}{17}i is \frac{771}{17}.

$$\frac{771}{17}$$