Solve -7+2i/7+3i=

$$\frac { - 7 + 2 i } { 7 + 3 i } =$$

Evaluate

$$-\frac{43}{58}+\frac{35}{58}i\approx -0.74137931+0.603448276i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 7-3i.

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

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

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

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

$$\frac{\left(-7+2i\right)\left(7-3i\right)}{58}$$

Multiply complex numbers -7+2i and 7-3i like you multiply binomials.

$$\frac{-7\times 7-7\times \left(-3i\right)+2i\times 7+2\left(-3\right)i^{2}}{58}$$

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

$$\frac{-7\times 7-7\times \left(-3i\right)+2i\times 7+2\left(-3\right)\left(-1\right)}{58}$$

Do the multiplications in -7\times 7-7\times \left(-3i\right)+2i\times 7+2\left(-3\right)\left(-1\right).

$$\frac{-49+21i+14i+6}{58}$$

Combine the real and imaginary parts in -49+21i+14i+6.

$$\frac{-49+6+\left(21+14\right)i}{58}$$

Do the additions in -49+6+\left(21+14\right)i.

$$\frac{-43+35i}{58}$$

Divide -43+35i by 58 to get -\frac{43}{58}+\frac{35}{58}i.

$$-\frac{43}{58}+\frac{35}{58}i$$

Real Part

$$-\frac{43}{58} = -0.7413793103448276$$

Solution Steps

Multiply both numerator and denominator of \frac{-7+2i}{7+3i} by the complex conjugate of the denominator, 7-3i.

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

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

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

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

$$Re(\frac{\left(-7+2i\right)\left(7-3i\right)}{58})$$

Multiply complex numbers -7+2i and 7-3i like you multiply binomials.

$$Re(\frac{-7\times 7-7\times \left(-3i\right)+2i\times 7+2\left(-3\right)i^{2}}{58})$$

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

$$Re(\frac{-7\times 7-7\times \left(-3i\right)+2i\times 7+2\left(-3\right)\left(-1\right)}{58})$$

Do the multiplications in -7\times 7-7\times \left(-3i\right)+2i\times 7+2\left(-3\right)\left(-1\right).

$$Re(\frac{-49+21i+14i+6}{58})$$

Combine the real and imaginary parts in -49+21i+14i+6.

$$Re(\frac{-49+6+\left(21+14\right)i}{58})$$

Do the additions in -49+6+\left(21+14\right)i.

$$Re(\frac{-43+35i}{58})$$

Divide -43+35i by 58 to get -\frac{43}{58}+\frac{35}{58}i.

$$Re(-\frac{43}{58}+\frac{35}{58}i)$$

The real part of -\frac{43}{58}+\frac{35}{58}i is -\frac{43}{58}.

$$-\frac{43}{58}$$