Solve 15x^{2}z^3+20x^2z^2=5xz^2(3xz+4x)

$$15 x ^ { 2 } z ^ { 3 } + 20 x ^ { 2 } z ^ { 2 } = 5 x z ^ { 2 } ( 3 x z + 4 x )$$

Solve for x (complex solution)

$$x\in \mathrm{C}$$

Solution Steps

Use the distributive property to multiply 5xz^{2} by 3xz+4x.

$$15x^{2}z^{3}+20x^{2}z^{2}=15x^{2}z^{3}+20x^{2}z^{2}$$

Subtract 15x^{2}z^{3} from both sides.

$$15x^{2}z^{3}+20x^{2}z^{2}-15x^{2}z^{3}=20x^{2}z^{2}$$

Combine 15x^{2}z^{3} and -15x^{2}z^{3} to get 0.

$$20x^{2}z^{2}=20x^{2}z^{2}$$

Subtract 20x^{2}z^{2} from both sides.

$$20x^{2}z^{2}-20x^{2}z^{2}=0$$

Combine 20x^{2}z^{2} and -20x^{2}z^{2} to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{C}$$

Solve for z (complex solution)

$$z\in \mathrm{C}$$

Solution Steps

Use the distributive property to multiply 5xz^{2} by 3xz+4x.

$$15x^{2}z^{3}+20x^{2}z^{2}=15x^{2}z^{3}+20x^{2}z^{2}$$

Subtract 15x^{2}z^{3} from both sides.

$$15x^{2}z^{3}+20x^{2}z^{2}-15x^{2}z^{3}=20x^{2}z^{2}$$

Combine 15x^{2}z^{3} and -15x^{2}z^{3} to get 0.

$$20x^{2}z^{2}=20x^{2}z^{2}$$

Subtract 20x^{2}z^{2} from both sides.

$$20x^{2}z^{2}-20x^{2}z^{2}=0$$

Combine 20x^{2}z^{2} and -20x^{2}z^{2} to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any z.

$$z\in \mathrm{C}$$

Solve for x

$$x\in \mathrm{R}$$

Solution Steps

Use the distributive property to multiply 5xz^{2} by 3xz+4x.

$$15x^{2}z^{3}+20x^{2}z^{2}=15x^{2}z^{3}+20x^{2}z^{2}$$

Subtract 15x^{2}z^{3} from both sides.

$$15x^{2}z^{3}+20x^{2}z^{2}-15x^{2}z^{3}=20x^{2}z^{2}$$

Combine 15x^{2}z^{3} and -15x^{2}z^{3} to get 0.

$$20x^{2}z^{2}=20x^{2}z^{2}$$

Subtract 20x^{2}z^{2} from both sides.

$$20x^{2}z^{2}-20x^{2}z^{2}=0$$

Combine 20x^{2}z^{2} and -20x^{2}z^{2} to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{R}$$

Solve for z

$$z\in \mathrm{R}$$

Solution Steps

Use the distributive property to multiply 5xz^{2} by 3xz+4x.

$$15x^{2}z^{3}+20x^{2}z^{2}=15x^{2}z^{3}+20x^{2}z^{2}$$

Subtract 15x^{2}z^{3} from both sides.

$$15x^{2}z^{3}+20x^{2}z^{2}-15x^{2}z^{3}=20x^{2}z^{2}$$

Combine 15x^{2}z^{3} and -15x^{2}z^{3} to get 0.

$$20x^{2}z^{2}=20x^{2}z^{2}$$

Subtract 20x^{2}z^{2} from both sides.

$$20x^{2}z^{2}-20x^{2}z^{2}=0$$

Combine 20x^{2}z^{2} and -20x^{2}z^{2} to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any z.

$$z\in \mathrm{R}$$