Solve 11+17i/-3-i

$$\frac { 11 + 17 i } { - 3 - i }$$

Evaluate

$$-5-4i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, -3+i.

$$\frac{\left(11+17i\right)\left(-3+i\right)}{\left(-3-i\right)\left(-3+i\right)}$$

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

$$\frac{\left(11+17i\right)\left(-3+i\right)}{\left(-3\right)^{2}-i^{2}}$$

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

$$\frac{\left(11+17i\right)\left(-3+i\right)}{10}$$

Multiply complex numbers 11+17i and -3+i like you multiply binomials.

$$\frac{11\left(-3\right)+11i+17i\left(-3\right)+17i^{2}}{10}$$

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

$$\frac{11\left(-3\right)+11i+17i\left(-3\right)+17\left(-1\right)}{10}$$

Do the multiplications in 11\left(-3\right)+11i+17i\left(-3\right)+17\left(-1\right).

$$\frac{-33+11i-51i-17}{10}$$

Combine the real and imaginary parts in -33+11i-51i-17.

$$\frac{-33-17+\left(11-51\right)i}{10}$$

Do the additions in -33-17+\left(11-51\right)i.

$$\frac{-50-40i}{10}$$

Divide -50-40i by 10 to get -5-4i.

$$-5-4i$$

Real Part

$$-5$$

Solution Steps

Multiply both numerator and denominator of \frac{11+17i}{-3-i} by the complex conjugate of the denominator, -3+i.

$$Re(\frac{\left(11+17i\right)\left(-3+i\right)}{\left(-3-i\right)\left(-3+i\right)})$$

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

$$Re(\frac{\left(11+17i\right)\left(-3+i\right)}{\left(-3\right)^{2}-i^{2}})$$

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

$$Re(\frac{\left(11+17i\right)\left(-3+i\right)}{10})$$

Multiply complex numbers 11+17i and -3+i like you multiply binomials.

$$Re(\frac{11\left(-3\right)+11i+17i\left(-3\right)+17i^{2}}{10})$$

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

$$Re(\frac{11\left(-3\right)+11i+17i\left(-3\right)+17\left(-1\right)}{10})$$

Do the multiplications in 11\left(-3\right)+11i+17i\left(-3\right)+17\left(-1\right).

$$Re(\frac{-33+11i-51i-17}{10})$$

Combine the real and imaginary parts in -33+11i-51i-17.

$$Re(\frac{-33-17+\left(11-51\right)i}{10})$$

Do the additions in -33-17+\left(11-51\right)i.

$$Re(\frac{-50-40i}{10})$$

Divide -50-40i by 10 to get -5-4i.

$$Re(-5-4i)$$

The real part of -5-4i is -5.

$$-5$$