Solve -17+20i/7-2i=

$$\frac { - 17 + 20 i } { 7 - 2 i } =$$

Evaluate

$$-3+2i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 7+2i.

$$\frac{\left(-17+20i\right)\left(7+2i\right)}{\left(7-2i\right)\left(7+2i\right)}$$

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

$$\frac{\left(-17+20i\right)\left(7+2i\right)}{7^{2}-2^{2}i^{2}}$$

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

$$\frac{\left(-17+20i\right)\left(7+2i\right)}{53}$$

Multiply complex numbers -17+20i and 7+2i like you multiply binomials.

$$\frac{-17\times 7-17\times \left(2i\right)+20i\times 7+20\times 2i^{2}}{53}$$

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

$$\frac{-17\times 7-17\times \left(2i\right)+20i\times 7+20\times 2\left(-1\right)}{53}$$

Do the multiplications in -17\times 7-17\times \left(2i\right)+20i\times 7+20\times 2\left(-1\right).

$$\frac{-119-34i+140i-40}{53}$$

Combine the real and imaginary parts in -119-34i+140i-40.

$$\frac{-119-40+\left(-34+140\right)i}{53}$$

Do the additions in -119-40+\left(-34+140\right)i.

$$\frac{-159+106i}{53}$$

Divide -159+106i by 53 to get -3+2i.

$$-3+2i$$

Real Part

$$-3$$

Solution Steps

Multiply both numerator and denominator of \frac{-17+20i}{7-2i} by the complex conjugate of the denominator, 7+2i.

$$Re(\frac{\left(-17+20i\right)\left(7+2i\right)}{\left(7-2i\right)\left(7+2i\right)})$$

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

$$Re(\frac{\left(-17+20i\right)\left(7+2i\right)}{7^{2}-2^{2}i^{2}})$$

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

$$Re(\frac{\left(-17+20i\right)\left(7+2i\right)}{53})$$

Multiply complex numbers -17+20i and 7+2i like you multiply binomials.

$$Re(\frac{-17\times 7-17\times \left(2i\right)+20i\times 7+20\times 2i^{2}}{53})$$

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

$$Re(\frac{-17\times 7-17\times \left(2i\right)+20i\times 7+20\times 2\left(-1\right)}{53})$$

Do the multiplications in -17\times 7-17\times \left(2i\right)+20i\times 7+20\times 2\left(-1\right).

$$Re(\frac{-119-34i+140i-40}{53})$$

Combine the real and imaginary parts in -119-34i+140i-40.

$$Re(\frac{-119-40+\left(-34+140\right)i}{53})$$

Do the additions in -119-40+\left(-34+140\right)i.

$$Re(\frac{-159+106i}{53})$$

Divide -159+106i by 53 to get -3+2i.

$$Re(-3+2i)$$

The real part of -3+2i is -3.

$$-3$$