Solve -5+4i/-4-i

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

Evaluate

$$\frac{16}{17}-\frac{21}{17}i\approx 0.941176471-1.235294118i$$

Solution Steps

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

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

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

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

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

$$\frac{\left(-5+4i\right)\left(-4+i\right)}{17}$$

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

$$\frac{-5\left(-4\right)-5i+4i\left(-4\right)+4i^{2}}{17}$$

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

$$\frac{-5\left(-4\right)-5i+4i\left(-4\right)+4\left(-1\right)}{17}$$

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

$$\frac{20-5i-16i-4}{17}$$

Combine the real and imaginary parts in 20-5i-16i-4.

$$\frac{20-4+\left(-5-16\right)i}{17}$$

Do the additions in 20-4+\left(-5-16\right)i.

$$\frac{16-21i}{17}$$

Divide 16-21i by 17 to get \frac{16}{17}-\frac{21}{17}i.

$$\frac{16}{17}-\frac{21}{17}i$$

Real Part

$$\frac{16}{17} = 0.9411764705882353$$

Solution Steps

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

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

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

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

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

$$Re(\frac{\left(-5+4i\right)\left(-4+i\right)}{17})$$

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

$$Re(\frac{-5\left(-4\right)-5i+4i\left(-4\right)+4i^{2}}{17})$$

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

$$Re(\frac{-5\left(-4\right)-5i+4i\left(-4\right)+4\left(-1\right)}{17})$$

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

$$Re(\frac{20-5i-16i-4}{17})$$

Combine the real and imaginary parts in 20-5i-16i-4.

$$Re(\frac{20-4+\left(-5-16\right)i}{17})$$

Do the additions in 20-4+\left(-5-16\right)i.

$$Re(\frac{16-21i}{17})$$

Divide 16-21i by 17 to get \frac{16}{17}-\frac{21}{17}i.

$$Re(\frac{16}{17}-\frac{21}{17}i)$$

The real part of \frac{16}{17}-\frac{21}{17}i is \frac{16}{17}.

$$\frac{16}{17}$$