Solve z=(3+2i)(5-3i)+7-5i

$$z = ( 3 + 2 i ) ( 5 - 3 i ) + 7 - 5 i$$

Solve for z

$$z=28-4i$$

Steps for Solving Linear Equation

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

$$z=3\times 5+3\times \left(-3i\right)+2i\times 5+2\left(-3\right)i^{2}+7-5i$$

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

$$z=3\times 5+3\times \left(-3i\right)+2i\times 5+2\left(-3\right)\left(-1\right)+7-5i$$

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

$$z=15-9i+10i+6+7-5i$$

Combine the real and imaginary parts in 15-9i+10i+6.

$$z=15+6+\left(-9+10\right)i+7-5i$$

Do the additions in 15+6+\left(-9+10\right)i.

$$z=21+i+7-5i$$

Combine the real and imaginary parts in 21+i+7-5i.

$$z=21+7+\left(1-5\right)i$$

Do the additions in 21+7+\left(1-5\right)i.

$$z=28-4i$$

Assign z

$$z≔28-4i$$