Solve -3-2i/5+2i

$$\frac { - 3 - 2 i } { 5 + 2 i }$$

Evaluate

$$-\frac{19}{29}-\frac{4}{29}i\approx -0.655172414-0.137931034i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 5-2i.

$$\frac{\left(-3-2i\right)\left(5-2i\right)}{\left(5+2i\right)\left(5-2i\right)}$$

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

$$\frac{\left(-3-2i\right)\left(5-2i\right)}{5^{2}-2^{2}i^{2}}$$

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

$$\frac{\left(-3-2i\right)\left(5-2i\right)}{29}$$

Multiply complex numbers -3-2i and 5-2i like you multiply binomials.

$$\frac{-3\times 5-3\times \left(-2i\right)-2i\times 5-2\left(-2\right)i^{2}}{29}$$

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

$$\frac{-3\times 5-3\times \left(-2i\right)-2i\times 5-2\left(-2\right)\left(-1\right)}{29}$$

Do the multiplications in -3\times 5-3\times \left(-2i\right)-2i\times 5-2\left(-2\right)\left(-1\right).

$$\frac{-15+6i-10i-4}{29}$$

Combine the real and imaginary parts in -15+6i-10i-4.

$$\frac{-15-4+\left(6-10\right)i}{29}$$

Do the additions in -15-4+\left(6-10\right)i.

$$\frac{-19-4i}{29}$$

Divide -19-4i by 29 to get -\frac{19}{29}-\frac{4}{29}i.

$$-\frac{19}{29}-\frac{4}{29}i$$

Real Part

$$-\frac{19}{29} = -0.6551724137931034$$

Solution Steps

Multiply both numerator and denominator of \frac{-3-2i}{5+2i} by the complex conjugate of the denominator, 5-2i.

$$Re(\frac{\left(-3-2i\right)\left(5-2i\right)}{\left(5+2i\right)\left(5-2i\right)})$$

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

$$Re(\frac{\left(-3-2i\right)\left(5-2i\right)}{5^{2}-2^{2}i^{2}})$$

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

$$Re(\frac{\left(-3-2i\right)\left(5-2i\right)}{29})$$

Multiply complex numbers -3-2i and 5-2i like you multiply binomials.

$$Re(\frac{-3\times 5-3\times \left(-2i\right)-2i\times 5-2\left(-2\right)i^{2}}{29})$$

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

$$Re(\frac{-3\times 5-3\times \left(-2i\right)-2i\times 5-2\left(-2\right)\left(-1\right)}{29})$$

Do the multiplications in -3\times 5-3\times \left(-2i\right)-2i\times 5-2\left(-2\right)\left(-1\right).

$$Re(\frac{-15+6i-10i-4}{29})$$

Combine the real and imaginary parts in -15+6i-10i-4.

$$Re(\frac{-15-4+\left(6-10\right)i}{29})$$

Do the additions in -15-4+\left(6-10\right)i.

$$Re(\frac{-19-4i}{29})$$

Divide -19-4i by 29 to get -\frac{19}{29}-\frac{4}{29}i.

$$Re(-\frac{19}{29}-\frac{4}{29}i)$$

The real part of -\frac{19}{29}-\frac{4}{29}i is -\frac{19}{29}.

$$-\frac{19}{29}$$