Solve -10+10i/4+2i

$$\frac { - 10 + 10 i } { 4 + 2 i }$$

Evaluate

$$-1+3i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 4-2i.

$$\frac{\left(-10+10i\right)\left(4-2i\right)}{\left(4+2i\right)\left(4-2i\right)}$$

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

$$\frac{\left(-10+10i\right)\left(4-2i\right)}{4^{2}-2^{2}i^{2}}$$

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

$$\frac{\left(-10+10i\right)\left(4-2i\right)}{20}$$

Multiply complex numbers -10+10i and 4-2i like you multiply binomials.

$$\frac{-10\times 4-10\times \left(-2i\right)+10i\times 4+10\left(-2\right)i^{2}}{20}$$

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

$$\frac{-10\times 4-10\times \left(-2i\right)+10i\times 4+10\left(-2\right)\left(-1\right)}{20}$$

Do the multiplications in -10\times 4-10\times \left(-2i\right)+10i\times 4+10\left(-2\right)\left(-1\right).

$$\frac{-40+20i+40i+20}{20}$$

Combine the real and imaginary parts in -40+20i+40i+20.

$$\frac{-40+20+\left(20+40\right)i}{20}$$

Do the additions in -40+20+\left(20+40\right)i.

$$\frac{-20+60i}{20}$$

Divide -20+60i by 20 to get -1+3i.

$$-1+3i$$

Real Part

$$-1$$

Solution Steps

Multiply both numerator and denominator of \frac{-10+10i}{4+2i} by the complex conjugate of the denominator, 4-2i.

$$Re(\frac{\left(-10+10i\right)\left(4-2i\right)}{\left(4+2i\right)\left(4-2i\right)})$$

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

$$Re(\frac{\left(-10+10i\right)\left(4-2i\right)}{4^{2}-2^{2}i^{2}})$$

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

$$Re(\frac{\left(-10+10i\right)\left(4-2i\right)}{20})$$

Multiply complex numbers -10+10i and 4-2i like you multiply binomials.

$$Re(\frac{-10\times 4-10\times \left(-2i\right)+10i\times 4+10\left(-2\right)i^{2}}{20})$$

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

$$Re(\frac{-10\times 4-10\times \left(-2i\right)+10i\times 4+10\left(-2\right)\left(-1\right)}{20})$$

Do the multiplications in -10\times 4-10\times \left(-2i\right)+10i\times 4+10\left(-2\right)\left(-1\right).

$$Re(\frac{-40+20i+40i+20}{20})$$

Combine the real and imaginary parts in -40+20i+40i+20.

$$Re(\frac{-40+20+\left(20+40\right)i}{20})$$

Do the additions in -40+20+\left(20+40\right)i.

$$Re(\frac{-20+60i}{20})$$

Divide -20+60i by 20 to get -1+3i.

$$Re(-1+3i)$$

The real part of -1+3i is -1.

$$-1$$