Solve -3+5i/-2+i

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

Evaluate

$$\frac{11}{5}-\frac{7}{5}i=2.2-1.4i$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

$$\frac{6+3i-10i+5}{5}$$

Combine the real and imaginary parts in 6+3i-10i+5.

$$\frac{6+5+\left(3-10\right)i}{5}$$

Do the additions in 6+5+\left(3-10\right)i.

$$\frac{11-7i}{5}$$

Divide 11-7i by 5 to get \frac{11}{5}-\frac{7}{5}i.

$$\frac{11}{5}-\frac{7}{5}i$$

Real Part

$$\frac{11}{5} = 2\frac{1}{5} = 2.2$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

$$Re(\frac{6+3i-10i+5}{5})$$

Combine the real and imaginary parts in 6+3i-10i+5.

$$Re(\frac{6+5+\left(3-10\right)i}{5})$$

Do the additions in 6+5+\left(3-10\right)i.

$$Re(\frac{11-7i}{5})$$

Divide 11-7i by 5 to get \frac{11}{5}-\frac{7}{5}i.

$$Re(\frac{11}{5}-\frac{7}{5}i)$$

The real part of \frac{11}{5}-\frac{7}{5}i is \frac{11}{5}.

$$\frac{11}{5}$$