Solve 9-19i/1-4i

$$\frac { 9 - 19 i } { 1 - 4 i }$$

Evaluate

$$5+i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 1+4i.

$$\frac{\left(9-19i\right)\left(1+4i\right)}{\left(1-4i\right)\left(1+4i\right)}$$

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

$$\frac{\left(9-19i\right)\left(1+4i\right)}{1^{2}-4^{2}i^{2}}$$

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

$$\frac{\left(9-19i\right)\left(1+4i\right)}{17}$$

Multiply complex numbers 9-19i and 1+4i like you multiply binomials.

$$\frac{9\times 1+9\times \left(4i\right)-19i-19\times 4i^{2}}{17}$$

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

$$\frac{9\times 1+9\times \left(4i\right)-19i-19\times 4\left(-1\right)}{17}$$

Do the multiplications in 9\times 1+9\times \left(4i\right)-19i-19\times 4\left(-1\right).

$$\frac{9+36i-19i+76}{17}$$

Combine the real and imaginary parts in 9+36i-19i+76.

$$\frac{9+76+\left(36-19\right)i}{17}$$

Do the additions in 9+76+\left(36-19\right)i.

$$\frac{85+17i}{17}$$

Divide 85+17i by 17 to get 5+i.

$$5+i$$

Real Part

$$5$$

Solution Steps

Multiply both numerator and denominator of \frac{9-19i}{1-4i} by the complex conjugate of the denominator, 1+4i.

$$Re(\frac{\left(9-19i\right)\left(1+4i\right)}{\left(1-4i\right)\left(1+4i\right)})$$

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

$$Re(\frac{\left(9-19i\right)\left(1+4i\right)}{1^{2}-4^{2}i^{2}})$$

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

$$Re(\frac{\left(9-19i\right)\left(1+4i\right)}{17})$$

Multiply complex numbers 9-19i and 1+4i like you multiply binomials.

$$Re(\frac{9\times 1+9\times \left(4i\right)-19i-19\times 4i^{2}}{17})$$

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

$$Re(\frac{9\times 1+9\times \left(4i\right)-19i-19\times 4\left(-1\right)}{17})$$

Do the multiplications in 9\times 1+9\times \left(4i\right)-19i-19\times 4\left(-1\right).

$$Re(\frac{9+36i-19i+76}{17})$$

Combine the real and imaginary parts in 9+36i-19i+76.

$$Re(\frac{9+76+\left(36-19\right)i}{17})$$

Do the additions in 9+76+\left(36-19\right)i.

$$Re(\frac{85+17i}{17})$$

Divide 85+17i by 17 to get 5+i.

$$Re(5+i)$$

The real part of 5+i is 5.

$$5$$