Solve (3+4texttt{i})(5-2texttt{i})

$$(3+4 \texttt{i} )(5-2 \texttt{i} )$$

Evaluate

$$23+14i$$

Solution Steps

Multiply complex numbers 3+4i and 5-2i like you multiply binomials.

$$3\times 5+3\times \left(-2i\right)+4i\times 5+4\left(-2\right)i^{2}$$

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

$$3\times 5+3\times \left(-2i\right)+4i\times 5+4\left(-2\right)\left(-1\right)$$

Do the multiplications.

$$15-6i+20i+8$$

Combine the real and imaginary parts.

$$15+8+\left(-6+20\right)i$$

Do the additions.

$$23+14i$$

Real Part

$$23$$

Solution Steps

Multiply complex numbers 3+4i and 5-2i like you multiply binomials.

$$Re(3\times 5+3\times \left(-2i\right)+4i\times 5+4\left(-2\right)i^{2})$$

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

$$Re(3\times 5+3\times \left(-2i\right)+4i\times 5+4\left(-2\right)\left(-1\right))$$

Do the multiplications in 3\times 5+3\times \left(-2i\right)+4i\times 5+4\left(-2\right)\left(-1\right).

$$Re(15-6i+20i+8)$$

Combine the real and imaginary parts in 15-6i+20i+8.

$$Re(15+8+\left(-6+20\right)i)$$

Do the additions in 15+8+\left(-6+20\right)i.

$$Re(23+14i)$$

The real part of 23+14i is 23.

$$23$$