Solve -3+9i/2+i

$$\frac { - 3 + 9 i } { 2 + i }$$

Evaluate

$$\frac{3}{5}+\frac{21}{5}i=0.6+4.2i$$

Solution Steps

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

$$\frac{\left(-3+9i\right)\left(2-i\right)}{\left(2+i\right)\left(2-i\right)}$$

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

$$\frac{\left(-3+9i\right)\left(2-i\right)}{2^{2}-i^{2}}$$

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

$$\frac{\left(-3+9i\right)\left(2-i\right)}{5}$$

Multiply complex numbers -3+9i and 2-i like you multiply binomials.

$$\frac{-3\times 2-3\left(-i\right)+9i\times 2+9\left(-1\right)i^{2}}{5}$$

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

$$\frac{-3\times 2-3\left(-i\right)+9i\times 2+9\left(-1\right)\left(-1\right)}{5}$$

Do the multiplications in -3\times 2-3\left(-i\right)+9i\times 2+9\left(-1\right)\left(-1\right).

$$\frac{-6+3i+18i+9}{5}$$

Combine the real and imaginary parts in -6+3i+18i+9.

$$\frac{-6+9+\left(3+18\right)i}{5}$$

Do the additions in -6+9+\left(3+18\right)i.

$$\frac{3+21i}{5}$$

Divide 3+21i by 5 to get \frac{3}{5}+\frac{21}{5}i.

$$\frac{3}{5}+\frac{21}{5}i$$

Real Part

$$\frac{3}{5} = 0.6$$

Solution Steps

Multiply both numerator and denominator of \frac{-3+9i}{2+i} by the complex conjugate of the denominator, 2-i.

$$Re(\frac{\left(-3+9i\right)\left(2-i\right)}{\left(2+i\right)\left(2-i\right)})$$

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

$$Re(\frac{\left(-3+9i\right)\left(2-i\right)}{2^{2}-i^{2}})$$

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

$$Re(\frac{\left(-3+9i\right)\left(2-i\right)}{5})$$

Multiply complex numbers -3+9i and 2-i like you multiply binomials.

$$Re(\frac{-3\times 2-3\left(-i\right)+9i\times 2+9\left(-1\right)i^{2}}{5})$$

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

$$Re(\frac{-3\times 2-3\left(-i\right)+9i\times 2+9\left(-1\right)\left(-1\right)}{5})$$

Do the multiplications in -3\times 2-3\left(-i\right)+9i\times 2+9\left(-1\right)\left(-1\right).

$$Re(\frac{-6+3i+18i+9}{5})$$

Combine the real and imaginary parts in -6+3i+18i+9.

$$Re(\frac{-6+9+\left(3+18\right)i}{5})$$

Do the additions in -6+9+\left(3+18\right)i.

$$Re(\frac{3+21i}{5})$$

Divide 3+21i by 5 to get \frac{3}{5}+\frac{21}{5}i.

$$Re(\frac{3}{5}+\frac{21}{5}i)$$

The real part of \frac{3}{5}+\frac{21}{5}i is \frac{3}{5}.

$$\frac{3}{5}$$