Solve -5-2i/-6+5i

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

Evaluate

$$\frac{20}{61}+\frac{37}{61}i\approx 0.327868852+0.606557377i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, -6-5i.

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

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

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

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

$$\frac{\left(-5-2i\right)\left(-6-5i\right)}{61}$$

Multiply complex numbers -5-2i and -6-5i like you multiply binomials.

$$\frac{-5\left(-6\right)-5\times \left(-5i\right)-2i\left(-6\right)-2\left(-5\right)i^{2}}{61}$$

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

$$\frac{-5\left(-6\right)-5\times \left(-5i\right)-2i\left(-6\right)-2\left(-5\right)\left(-1\right)}{61}$$

Do the multiplications in -5\left(-6\right)-5\times \left(-5i\right)-2i\left(-6\right)-2\left(-5\right)\left(-1\right).

$$\frac{30+25i+12i-10}{61}$$

Combine the real and imaginary parts in 30+25i+12i-10.

$$\frac{30-10+\left(25+12\right)i}{61}$$

Do the additions in 30-10+\left(25+12\right)i.

$$\frac{20+37i}{61}$$

Divide 20+37i by 61 to get \frac{20}{61}+\frac{37}{61}i.

$$\frac{20}{61}+\frac{37}{61}i$$

Real Part

$$\frac{20}{61} = 0.32786885245901637$$

Solution Steps

Multiply both numerator and denominator of \frac{-5-2i}{-6+5i} by the complex conjugate of the denominator, -6-5i.

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

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

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

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

$$Re(\frac{\left(-5-2i\right)\left(-6-5i\right)}{61})$$

Multiply complex numbers -5-2i and -6-5i like you multiply binomials.

$$Re(\frac{-5\left(-6\right)-5\times \left(-5i\right)-2i\left(-6\right)-2\left(-5\right)i^{2}}{61})$$

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

$$Re(\frac{-5\left(-6\right)-5\times \left(-5i\right)-2i\left(-6\right)-2\left(-5\right)\left(-1\right)}{61})$$

Do the multiplications in -5\left(-6\right)-5\times \left(-5i\right)-2i\left(-6\right)-2\left(-5\right)\left(-1\right).

$$Re(\frac{30+25i+12i-10}{61})$$

Combine the real and imaginary parts in 30+25i+12i-10.

$$Re(\frac{30-10+\left(25+12\right)i}{61})$$

Do the additions in 30-10+\left(25+12\right)i.

$$Re(\frac{20+37i}{61})$$

Divide 20+37i by 61 to get \frac{20}{61}+\frac{37}{61}i.

$$Re(\frac{20}{61}+\frac{37}{61}i)$$

The real part of \frac{20}{61}+\frac{37}{61}i is \frac{20}{61}.

$$\frac{20}{61}$$