Solve (u-vi)*(v-vi)-(v+vi)*(v+vi)/(u+vi)*(v-vi)

$$\frac { ( u - v i ) \cdot ( v - v i ) - ( v + v i ) \cdot ( v + v i ) } { ( u + v i ) \cdot ( v - v i ) }$$

Evaluate

$$\frac{iu+\left(2+i\right)v}{iu-v}$$

Solution Steps

Multiply v+vi and v+vi to get \left(v+vi\right)^{2}.

$$\frac{\left(u-vi\right)\left(v-vi\right)-\left(v+vi\right)^{2}}{\left(u+vi\right)\left(v-vi\right)}$$

Combine v and -vi to get \left(1-i\right)v.

$$\frac{\left(u-vi\right)\left(1-i\right)v-\left(v+vi\right)^{2}}{\left(u+vi\right)\left(v-vi\right)}$$

Combine v and vi to get \left(1+i\right)v.

$$\frac{\left(u-vi\right)\left(1-i\right)v-\left(\left(1+i\right)v\right)^{2}}{\left(u+vi\right)\left(v-vi\right)}$$

Expand \left(\left(1+i\right)v\right)^{2}.

$$\frac{\left(u-vi\right)\left(1-i\right)v-\left(1+i\right)^{2}v^{2}}{\left(u+vi\right)\left(v-vi\right)}$$

Calculate 1+i to the power of 2 and get 2i.

$$\frac{\left(u-vi\right)\left(1-i\right)v-2iv^{2}}{\left(u+vi\right)\left(v-vi\right)}$$

Combine v and -vi to get \left(1-i\right)v.

$$\frac{\left(u-vi\right)\left(1-i\right)v-2iv^{2}}{\left(u+vi\right)\left(1-i\right)v}$$

Use the distributive property to multiply u+vi by 1-i.

$$\frac{\left(u-vi\right)\left(1-i\right)v-2iv^{2}}{\left(\left(1-i\right)u+\left(1-i\right)vi\right)v}$$

Multiply 1-i times i.

$$\frac{\left(u-vi\right)\left(1-i\right)v-2iv^{2}}{\left(\left(1-i\right)u+\left(i-i^{2}\right)v\right)v}$$

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

$$\frac{\left(u-vi\right)\left(1-i\right)v-2iv^{2}}{\left(\left(1-i\right)u+\left(i-\left(-1\right)\right)v\right)v}$$

Do the multiplications in i-\left(-1\right). Reorder the terms.

$$\frac{\left(u-vi\right)\left(1-i\right)v-2iv^{2}}{\left(\left(1-i\right)u+\left(1+i\right)v\right)v}$$

Use the distributive property to multiply \left(1-i\right)u+\left(1+i\right)v by v.

$$\frac{\left(u-vi\right)\left(1-i\right)v-2iv^{2}}{\left(1-i\right)uv+\left(1+i\right)v^{2}}$$

Multiply -1 and i to get -i.

$$\frac{\left(u-iv\right)\left(1-i\right)v-2iv^{2}}{\left(1-i\right)uv+\left(1+i\right)v^{2}}$$

Use the distributive property to multiply u-iv by 1-i.

$$\frac{\left(\left(1-i\right)u+\left(-1-i\right)v\right)v-2iv^{2}}{\left(1-i\right)uv+\left(1+i\right)v^{2}}$$

Use the distributive property to multiply \left(1-i\right)u+\left(-1-i\right)v by v.

$$\frac{\left(1-i\right)uv+\left(-1-i\right)v^{2}-2iv^{2}}{\left(1-i\right)uv+\left(1+i\right)v^{2}}$$

Combine \left(-1-i\right)v^{2} and -2iv^{2} to get \left(-1-3i\right)v^{2}.

$$\frac{\left(1-i\right)uv+\left(-1-3i\right)v^{2}}{\left(1-i\right)uv+\left(1+i\right)v^{2}}$$

Expand

$$\frac{u+\left(1-2i\right)v}{u+iv}$$

Solution Steps

Multiply v+vi and v+vi to get \left(v+vi\right)^{2}.

$$\frac{\left(u-vi\right)\left(v-vi\right)-\left(v+vi\right)^{2}}{\left(u+vi\right)\left(v-vi\right)}$$

Combine v and -vi to get \left(1-i\right)v.

$$\frac{\left(u-vi\right)\left(1-i\right)v-\left(v+vi\right)^{2}}{\left(u+vi\right)\left(v-vi\right)}$$

Combine v and vi to get \left(1+i\right)v.

$$\frac{\left(u-vi\right)\left(1-i\right)v-\left(\left(1+i\right)v\right)^{2}}{\left(u+vi\right)\left(v-vi\right)}$$

Expand \left(\left(1+i\right)v\right)^{2}.

$$\frac{\left(u-vi\right)\left(1-i\right)v-\left(1+i\right)^{2}v^{2}}{\left(u+vi\right)\left(v-vi\right)}$$

Calculate 1+i to the power of 2 and get 2i.

$$\frac{\left(u-vi\right)\left(1-i\right)v-2iv^{2}}{\left(u+vi\right)\left(v-vi\right)}$$

Combine v and -vi to get \left(1-i\right)v.

$$\frac{\left(u-vi\right)\left(1-i\right)v-2iv^{2}}{\left(u+vi\right)\left(1-i\right)v}$$

Use the distributive property to multiply u+vi by 1-i.

$$\frac{\left(u-vi\right)\left(1-i\right)v-2iv^{2}}{\left(\left(1-i\right)u+\left(1-i\right)vi\right)v}$$

Multiply 1-i times i.

$$\frac{\left(u-vi\right)\left(1-i\right)v-2iv^{2}}{\left(\left(1-i\right)u+\left(i-i^{2}\right)v\right)v}$$

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

$$\frac{\left(u-vi\right)\left(1-i\right)v-2iv^{2}}{\left(\left(1-i\right)u+\left(i-\left(-1\right)\right)v\right)v}$$

Do the multiplications in i-\left(-1\right). Reorder the terms.

$$\frac{\left(u-vi\right)\left(1-i\right)v-2iv^{2}}{\left(\left(1-i\right)u+\left(1+i\right)v\right)v}$$

Use the distributive property to multiply \left(1-i\right)u+\left(1+i\right)v by v.

$$\frac{\left(u-vi\right)\left(1-i\right)v-2iv^{2}}{\left(1-i\right)uv+\left(1+i\right)v^{2}}$$

Multiply -1 and i to get -i.

$$\frac{\left(u-iv\right)\left(1-i\right)v-2iv^{2}}{\left(1-i\right)uv+\left(1+i\right)v^{2}}$$

Use the distributive property to multiply u-iv by 1-i.

$$\frac{\left(\left(1-i\right)u+\left(-1-i\right)v\right)v-2iv^{2}}{\left(1-i\right)uv+\left(1+i\right)v^{2}}$$

Use the distributive property to multiply \left(1-i\right)u+\left(-1-i\right)v by v.

$$\frac{\left(1-i\right)uv+\left(-1-i\right)v^{2}-2iv^{2}}{\left(1-i\right)uv+\left(1+i\right)v^{2}}$$

Combine \left(-1-i\right)v^{2} and -2iv^{2} to get \left(-1-3i\right)v^{2}.

$$\frac{\left(1-i\right)uv+\left(-1-3i\right)v^{2}}{\left(1-i\right)uv+\left(1+i\right)v^{2}}$$