Solve -10+6i/3+5i

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

Evaluate

$$2i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 3-5i.

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

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

$$\frac{\left(-10+6i\right)\left(3-5i\right)}{3^{2}-5^{2}i^{2}}$$

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

$$\frac{\left(-10+6i\right)\left(3-5i\right)}{34}$$

Multiply complex numbers -10+6i and 3-5i like you multiply binomials.

$$\frac{-10\times 3-10\times \left(-5i\right)+6i\times 3+6\left(-5\right)i^{2}}{34}$$

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

$$\frac{-10\times 3-10\times \left(-5i\right)+6i\times 3+6\left(-5\right)\left(-1\right)}{34}$$

Do the multiplications in -10\times 3-10\times \left(-5i\right)+6i\times 3+6\left(-5\right)\left(-1\right).

$$\frac{-30+50i+18i+30}{34}$$

Combine the real and imaginary parts in -30+50i+18i+30.

$$\frac{-30+30+\left(50+18\right)i}{34}$$

Do the additions in -30+30+\left(50+18\right)i.

$$\frac{68i}{34}$$

Divide 68i by 34 to get 2i.

$$2i$$

Real Part

$$0$$

Solution Steps

Multiply both numerator and denominator of \frac{-10+6i}{3+5i} by the complex conjugate of the denominator, 3-5i.

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

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

$$Re(\frac{\left(-10+6i\right)\left(3-5i\right)}{3^{2}-5^{2}i^{2}})$$

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

$$Re(\frac{\left(-10+6i\right)\left(3-5i\right)}{34})$$

Multiply complex numbers -10+6i and 3-5i like you multiply binomials.

$$Re(\frac{-10\times 3-10\times \left(-5i\right)+6i\times 3+6\left(-5\right)i^{2}}{34})$$

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

$$Re(\frac{-10\times 3-10\times \left(-5i\right)+6i\times 3+6\left(-5\right)\left(-1\right)}{34})$$

Do the multiplications in -10\times 3-10\times \left(-5i\right)+6i\times 3+6\left(-5\right)\left(-1\right).

$$Re(\frac{-30+50i+18i+30}{34})$$

Combine the real and imaginary parts in -30+50i+18i+30.

$$Re(\frac{-30+30+\left(50+18\right)i}{34})$$

Do the additions in -30+30+\left(50+18\right)i.

$$Re(\frac{68i}{34})$$

Divide 68i by 34 to get 2i.

$$Re(2i)$$

The real part of 2i is 0.

$$0$$