Solve -7+6i/10+5i

$$\frac { - 7 + 6 i } { 10 + 5 i }$$

Evaluate

$$-\frac{8}{25}+\frac{19}{25}i=-0.32+0.76i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 10-5i.

$$\frac{\left(-7+6i\right)\left(10-5i\right)}{\left(10+5i\right)\left(10-5i\right)}$$

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

$$\frac{\left(-7+6i\right)\left(10-5i\right)}{10^{2}-5^{2}i^{2}}$$

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

$$\frac{\left(-7+6i\right)\left(10-5i\right)}{125}$$

Multiply complex numbers -7+6i and 10-5i like you multiply binomials.

$$\frac{-7\times 10-7\times \left(-5i\right)+6i\times 10+6\left(-5\right)i^{2}}{125}$$

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

$$\frac{-7\times 10-7\times \left(-5i\right)+6i\times 10+6\left(-5\right)\left(-1\right)}{125}$$

Do the multiplications in -7\times 10-7\times \left(-5i\right)+6i\times 10+6\left(-5\right)\left(-1\right).

$$\frac{-70+35i+60i+30}{125}$$

Combine the real and imaginary parts in -70+35i+60i+30.

$$\frac{-70+30+\left(35+60\right)i}{125}$$

Do the additions in -70+30+\left(35+60\right)i.

$$\frac{-40+95i}{125}$$

Divide -40+95i by 125 to get -\frac{8}{25}+\frac{19}{25}i.

$$-\frac{8}{25}+\frac{19}{25}i$$

Real Part

$$-\frac{8}{25} = -0.32$$

Solution Steps

Multiply both numerator and denominator of \frac{-7+6i}{10+5i} by the complex conjugate of the denominator, 10-5i.

$$Re(\frac{\left(-7+6i\right)\left(10-5i\right)}{\left(10+5i\right)\left(10-5i\right)})$$

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

$$Re(\frac{\left(-7+6i\right)\left(10-5i\right)}{10^{2}-5^{2}i^{2}})$$

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

$$Re(\frac{\left(-7+6i\right)\left(10-5i\right)}{125})$$

Multiply complex numbers -7+6i and 10-5i like you multiply binomials.

$$Re(\frac{-7\times 10-7\times \left(-5i\right)+6i\times 10+6\left(-5\right)i^{2}}{125})$$

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

$$Re(\frac{-7\times 10-7\times \left(-5i\right)+6i\times 10+6\left(-5\right)\left(-1\right)}{125})$$

Do the multiplications in -7\times 10-7\times \left(-5i\right)+6i\times 10+6\left(-5\right)\left(-1\right).

$$Re(\frac{-70+35i+60i+30}{125})$$

Combine the real and imaginary parts in -70+35i+60i+30.

$$Re(\frac{-70+30+\left(35+60\right)i}{125})$$

Do the additions in -70+30+\left(35+60\right)i.

$$Re(\frac{-40+95i}{125})$$

Divide -40+95i by 125 to get -\frac{8}{25}+\frac{19}{25}i.

$$Re(-\frac{8}{25}+\frac{19}{25}i)$$

The real part of -\frac{8}{25}+\frac{19}{25}i is -\frac{8}{25}.

$$-\frac{8}{25}$$