Solve left(x-1right)^3+3/2(x(x+6)+1)=left(2+xright)^3

$${ \left(x-1 \right) }^{ 3 } + \frac{ 3 }{ 2 } (x(x+6)+1)= { \left(2+x \right) }^{ 3 }$$

Solve for x (complex solution)

$$x=i$$

$$x=-i$$

Steps by Finding Square Root

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x-1\right)^{3}.

$$x^{3}-3x^{2}+3x-1+\frac{3}{2}\left(x\left(x+6\right)+1\right)=\left(2+x\right)^{3}$$

Use the distributive property to multiply x by x+6.

$$x^{3}-3x^{2}+3x-1+\frac{3}{2}\left(x^{2}+6x+1\right)=\left(2+x\right)^{3}$$

Use the distributive property to multiply \frac{3}{2} by x^{2}+6x+1.

$$x^{3}-3x^{2}+3x-1+\frac{3}{2}x^{2}+9x+\frac{3}{2}=\left(2+x\right)^{3}$$

Combine -3x^{2} and \frac{3}{2}x^{2} to get -\frac{3}{2}x^{2}.

$$x^{3}-\frac{3}{2}x^{2}+3x-1+9x+\frac{3}{2}=\left(2+x\right)^{3}$$

Combine 3x and 9x to get 12x.

$$x^{3}-\frac{3}{2}x^{2}+12x-1+\frac{3}{2}=\left(2+x\right)^{3}$$

Add -1 and \frac{3}{2} to get \frac{1}{2}.

$$x^{3}-\frac{3}{2}x^{2}+12x+\frac{1}{2}=\left(2+x\right)^{3}$$

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

$$x^{3}-\frac{3}{2}x^{2}+12x+\frac{1}{2}=8+12x+6x^{2}+x^{3}$$

Subtract 12x from both sides.

$$x^{3}-\frac{3}{2}x^{2}+12x+\frac{1}{2}-12x=8+6x^{2}+x^{3}$$

Combine 12x and -12x to get 0.

$$x^{3}-\frac{3}{2}x^{2}+\frac{1}{2}=8+6x^{2}+x^{3}$$

Subtract 6x^{2} from both sides.

$$x^{3}-\frac{3}{2}x^{2}+\frac{1}{2}-6x^{2}=8+x^{3}$$

Combine -\frac{3}{2}x^{2} and -6x^{2} to get -\frac{15}{2}x^{2}.

$$x^{3}-\frac{15}{2}x^{2}+\frac{1}{2}=8+x^{3}$$

Subtract x^{3} from both sides.

$$x^{3}-\frac{15}{2}x^{2}+\frac{1}{2}-x^{3}=8$$

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

$$-\frac{15}{2}x^{2}+\frac{1}{2}=8$$

Subtract \frac{1}{2} from both sides.

$$-\frac{15}{2}x^{2}=8-\frac{1}{2}$$

Subtract \frac{1}{2} from 8 to get \frac{15}{2}.

$$-\frac{15}{2}x^{2}=\frac{15}{2}$$

Multiply both sides by -\frac{2}{15}, the reciprocal of -\frac{15}{2}.

$$x^{2}=\frac{15}{2}\left(-\frac{2}{15}\right)$$

Multiply \frac{15}{2} and -\frac{2}{15} to get -1.

$$x^{2}=-1$$

The equation is now solved.

$$x=i x=-i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x-1\right)^{3}.

$$x^{3}-3x^{2}+3x-1+\frac{3}{2}\left(x\left(x+6\right)+1\right)=\left(2+x\right)^{3}$$

Use the distributive property to multiply x by x+6.

$$x^{3}-3x^{2}+3x-1+\frac{3}{2}\left(x^{2}+6x+1\right)=\left(2+x\right)^{3}$$

Use the distributive property to multiply \frac{3}{2} by x^{2}+6x+1.

$$x^{3}-3x^{2}+3x-1+\frac{3}{2}x^{2}+9x+\frac{3}{2}=\left(2+x\right)^{3}$$

Combine -3x^{2} and \frac{3}{2}x^{2} to get -\frac{3}{2}x^{2}.

$$x^{3}-\frac{3}{2}x^{2}+3x-1+9x+\frac{3}{2}=\left(2+x\right)^{3}$$

Combine 3x and 9x to get 12x.

$$x^{3}-\frac{3}{2}x^{2}+12x-1+\frac{3}{2}=\left(2+x\right)^{3}$$

Add -1 and \frac{3}{2} to get \frac{1}{2}.

$$x^{3}-\frac{3}{2}x^{2}+12x+\frac{1}{2}=\left(2+x\right)^{3}$$

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

$$x^{3}-\frac{3}{2}x^{2}+12x+\frac{1}{2}=8+12x+6x^{2}+x^{3}$$

Subtract 8 from both sides.

$$x^{3}-\frac{3}{2}x^{2}+12x+\frac{1}{2}-8=12x+6x^{2}+x^{3}$$

Subtract 8 from \frac{1}{2} to get -\frac{15}{2}.

$$x^{3}-\frac{3}{2}x^{2}+12x-\frac{15}{2}=12x+6x^{2}+x^{3}$$

Subtract 12x from both sides.

$$x^{3}-\frac{3}{2}x^{2}+12x-\frac{15}{2}-12x=6x^{2}+x^{3}$$

Combine 12x and -12x to get 0.

$$x^{3}-\frac{3}{2}x^{2}-\frac{15}{2}=6x^{2}+x^{3}$$

Subtract 6x^{2} from both sides.

$$x^{3}-\frac{3}{2}x^{2}-\frac{15}{2}-6x^{2}=x^{3}$$

Combine -\frac{3}{2}x^{2} and -6x^{2} to get -\frac{15}{2}x^{2}.

$$x^{3}-\frac{15}{2}x^{2}-\frac{15}{2}=x^{3}$$

Subtract x^{3} from both sides.

$$x^{3}-\frac{15}{2}x^{2}-\frac{15}{2}-x^{3}=0$$

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

$$-\frac{15}{2}x^{2}-\frac{15}{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{15}{2} for a, 0 for b, and -\frac{15}{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-\frac{15}{2}\right)\left(-\frac{15}{2}\right)}}{2\left(-\frac{15}{2}\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-\frac{15}{2}\right)\left(-\frac{15}{2}\right)}}{2\left(-\frac{15}{2}\right)}$$

Multiply -4 times -\frac{15}{2}.

$$x=\frac{0±\sqrt{30\left(-\frac{15}{2}\right)}}{2\left(-\frac{15}{2}\right)}$$

Multiply 30 times -\frac{15}{2}.

$$x=\frac{0±\sqrt{-225}}{2\left(-\frac{15}{2}\right)}$$

Take the square root of -225.

$$x=\frac{0±15i}{2\left(-\frac{15}{2}\right)}$$

Multiply 2 times -\frac{15}{2}.

$$x=\frac{0±15i}{-15}$$

Now solve the equation x=\frac{0±15i}{-15} when ± is plus.

$$x=-i$$

Now solve the equation x=\frac{0±15i}{-15} when ± is minus.

$$x=i$$

The equation is now solved.

$$x=-i x=i$$