Solve 5+2/-6+3i

$$\frac { 5 + 2 } { - 6 + 3 i }$$

Evaluate

$$-\frac{14}{15}-\frac{7}{15}i\approx -0.933333333-0.466666667i$$

Solution Steps

Add 5 and 2 to get 7.

$$\frac{7}{-6+3i}$$

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

$$\frac{7\left(-6-3i\right)}{\left(-6+3i\right)\left(-6-3i\right)}$$

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

$$\frac{7\left(-6-3i\right)}{\left(-6\right)^{2}-3^{2}i^{2}}$$

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

$$\frac{7\left(-6-3i\right)}{45}$$

Multiply 7 times -6-3i.

$$\frac{7\left(-6\right)+7\times \left(-3i\right)}{45}$$

Do the multiplications in 7\left(-6\right)+7\times \left(-3i\right).

$$\frac{-42-21i}{45}$$

Divide -42-21i by 45 to get -\frac{14}{15}-\frac{7}{15}i.

$$-\frac{14}{15}-\frac{7}{15}i$$

Real Part

$$-\frac{14}{15} = -0.9333333333333333$$

Solution Steps

Add 5 and 2 to get 7.

$$Re(\frac{7}{-6+3i})$$

Multiply both numerator and denominator of \frac{7}{-6+3i} by the complex conjugate of the denominator, -6-3i.

$$Re(\frac{7\left(-6-3i\right)}{\left(-6+3i\right)\left(-6-3i\right)})$$

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

$$Re(\frac{7\left(-6-3i\right)}{\left(-6\right)^{2}-3^{2}i^{2}})$$

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

$$Re(\frac{7\left(-6-3i\right)}{45})$$

Multiply 7 times -6-3i.

$$Re(\frac{7\left(-6\right)+7\times \left(-3i\right)}{45})$$

Do the multiplications in 7\left(-6\right)+7\times \left(-3i\right).

$$Re(\frac{-42-21i}{45})$$

Divide -42-21i by 45 to get -\frac{14}{15}-\frac{7}{15}i.

$$Re(-\frac{14}{15}-\frac{7}{15}i)$$

The real part of -\frac{14}{15}-\frac{7}{15}i is -\frac{14}{15}.

$$-\frac{14}{15}$$