Solve x^3-6x^2+12x-8=(x-2)^3

$$x ^ { 3 } - 6 x ^ { 2 } + 12 x - 8 = ( x - 2 ) ^ { 3 }$$

Solve for x (complex solution)

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

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x-2\right)^{3}.

$$x^{3}-6x^{2}+12x-8=x^{3}-6x^{2}+12x-8$$

Subtract x^{3} from both sides.

$$x^{3}-6x^{2}+12x-8-x^{3}=-6x^{2}+12x-8$$

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

$$-6x^{2}+12x-8=-6x^{2}+12x-8$$

Add 6x^{2} to both sides.

$$-6x^{2}+12x-8+6x^{2}=12x-8$$

Combine -6x^{2} and 6x^{2} to get 0.

$$12x-8=12x-8$$

Subtract 12x from both sides.

$$12x-8-12x=-8$$

Combine 12x and -12x to get 0.

$$-8=-8$$

Compare -8 and -8.

$$\text{true}$$

This is true for any x.

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

Solve for x

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

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x-2\right)^{3}.

$$x^{3}-6x^{2}+12x-8=x^{3}-6x^{2}+12x-8$$

Subtract x^{3} from both sides.

$$x^{3}-6x^{2}+12x-8-x^{3}=-6x^{2}+12x-8$$

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

$$-6x^{2}+12x-8=-6x^{2}+12x-8$$

Add 6x^{2} to both sides.

$$-6x^{2}+12x-8+6x^{2}=12x-8$$

Combine -6x^{2} and 6x^{2} to get 0.

$$12x-8=12x-8$$

Subtract 12x from both sides.

$$12x-8-12x=-8$$

Combine 12x and -12x to get 0.

$$-8=-8$$

Compare -8 and -8.

$$\text{true}$$

This is true for any x.

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