Solve (58-69i)div(-7-4i)

$$( 58 - 69 i ) \div ( - 7 - 4 i )$$

Evaluate

$$-2+11i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, -7+4i.

$$\frac{\left(58-69i\right)\left(-7+4i\right)}{\left(-7-4i\right)\left(-7+4i\right)}$$

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

$$\frac{\left(58-69i\right)\left(-7+4i\right)}{\left(-7\right)^{2}-4^{2}i^{2}}$$

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

$$\frac{\left(58-69i\right)\left(-7+4i\right)}{65}$$

Multiply complex numbers 58-69i and -7+4i like you multiply binomials.

$$\frac{58\left(-7\right)+58\times \left(4i\right)-69i\left(-7\right)-69\times 4i^{2}}{65}$$

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

$$\frac{58\left(-7\right)+58\times \left(4i\right)-69i\left(-7\right)-69\times 4\left(-1\right)}{65}$$

Do the multiplications in 58\left(-7\right)+58\times \left(4i\right)-69i\left(-7\right)-69\times 4\left(-1\right).

$$\frac{-406+232i+483i+276}{65}$$

Combine the real and imaginary parts in -406+232i+483i+276.

$$\frac{-406+276+\left(232+483\right)i}{65}$$

Do the additions in -406+276+\left(232+483\right)i.

$$\frac{-130+715i}{65}$$

Divide -130+715i by 65 to get -2+11i.

$$-2+11i$$

Real Part

$$-2$$

Solution Steps

Multiply both numerator and denominator of \frac{58-69i}{-7-4i} by the complex conjugate of the denominator, -7+4i.

$$Re(\frac{\left(58-69i\right)\left(-7+4i\right)}{\left(-7-4i\right)\left(-7+4i\right)})$$

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

$$Re(\frac{\left(58-69i\right)\left(-7+4i\right)}{\left(-7\right)^{2}-4^{2}i^{2}})$$

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

$$Re(\frac{\left(58-69i\right)\left(-7+4i\right)}{65})$$

Multiply complex numbers 58-69i and -7+4i like you multiply binomials.

$$Re(\frac{58\left(-7\right)+58\times \left(4i\right)-69i\left(-7\right)-69\times 4i^{2}}{65})$$

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

$$Re(\frac{58\left(-7\right)+58\times \left(4i\right)-69i\left(-7\right)-69\times 4\left(-1\right)}{65})$$

Do the multiplications in 58\left(-7\right)+58\times \left(4i\right)-69i\left(-7\right)-69\times 4\left(-1\right).

$$Re(\frac{-406+232i+483i+276}{65})$$

Combine the real and imaginary parts in -406+232i+483i+276.

$$Re(\frac{-406+276+\left(232+483\right)i}{65})$$

Do the additions in -406+276+\left(232+483\right)i.

$$Re(\frac{-130+715i}{65})$$

Divide -130+715i by 65 to get -2+11i.

$$Re(-2+11i)$$

The real part of -2+11i is -2.

$$-2$$