Solve 5+5i/-6-3i

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

Evaluate

$$-1-\frac{1}{3}i\approx -1-0.333333333i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, -6+3i.

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

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

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

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

$$\frac{\left(5+5i\right)\left(-6+3i\right)}{45}$$

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

$$\frac{5\left(-6\right)+5\times \left(3i\right)+5i\left(-6\right)+5\times 3i^{2}}{45}$$

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

$$\frac{5\left(-6\right)+5\times \left(3i\right)+5i\left(-6\right)+5\times 3\left(-1\right)}{45}$$

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

$$\frac{-30+15i-30i-15}{45}$$

Combine the real and imaginary parts in -30+15i-30i-15.

$$\frac{-30-15+\left(15-30\right)i}{45}$$

Do the additions in -30-15+\left(15-30\right)i.

$$\frac{-45-15i}{45}$$

Divide -45-15i by 45 to get -1-\frac{1}{3}i.

$$-1-\frac{1}{3}i$$

Real Part

$$-1$$

Solution Steps

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

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

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

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

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

$$Re(\frac{\left(5+5i\right)\left(-6+3i\right)}{45})$$

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

$$Re(\frac{5\left(-6\right)+5\times \left(3i\right)+5i\left(-6\right)+5\times 3i^{2}}{45})$$

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

$$Re(\frac{5\left(-6\right)+5\times \left(3i\right)+5i\left(-6\right)+5\times 3\left(-1\right)}{45})$$

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

$$Re(\frac{-30+15i-30i-15}{45})$$

Combine the real and imaginary parts in -30+15i-30i-15.

$$Re(\frac{-30-15+\left(15-30\right)i}{45})$$

Do the additions in -30-15+\left(15-30\right)i.

$$Re(\frac{-45-15i}{45})$$

Divide -45-15i by 45 to get -1-\frac{1}{3}i.

$$Re(-1-\frac{1}{3}i)$$

The real part of -1-\frac{1}{3}i is -1.

$$-1$$