Solve (x-1/3)^3-(2/5x-1)(2-x)-x(2/5x+3)=x^2(x-1)-1/3(2-x)

$$( x - \frac { 1 } { 3 } ) ^ { 3 } - ( \frac { 2 } { 5 } x - 1 ) ( 2 - x ) - x ( \frac { 2 } { 5 } x + 3 ) = x ^ { 2 } ( x - 1 ) - \frac { 1 } { 3 } ( 2 - x )$$

Solve for x

$$x=\frac{355}{648}\approx 0.547839506$$

Steps for Solving Linear Equation

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

$$x^{3}-x^{2}+\frac{1}{3}x-\frac{1}{27}-\left(\frac{2}{5}x-1\right)\left(2-x\right)-x\left(\frac{2}{5}x+3\right)=x^{2}\left(x-1\right)-\frac{1}{3}\left(2-x\right)$$

Use the distributive property to multiply \frac{2}{5}x-1 by 2-x and combine like terms.

$$x^{3}-x^{2}+\frac{1}{3}x-\frac{1}{27}-\left(\frac{9}{5}x-\frac{2}{5}x^{2}-2\right)-x\left(\frac{2}{5}x+3\right)=x^{2}\left(x-1\right)-\frac{1}{3}\left(2-x\right)$$

To find the opposite of \frac{9}{5}x-\frac{2}{5}x^{2}-2, find the opposite of each term.

$$x^{3}-x^{2}+\frac{1}{3}x-\frac{1}{27}-\frac{9}{5}x+\frac{2}{5}x^{2}+2-x\left(\frac{2}{5}x+3\right)=x^{2}\left(x-1\right)-\frac{1}{3}\left(2-x\right)$$

Combine \frac{1}{3}x and -\frac{9}{5}x to get -\frac{22}{15}x.

$$x^{3}-x^{2}-\frac{22}{15}x-\frac{1}{27}+\frac{2}{5}x^{2}+2-x\left(\frac{2}{5}x+3\right)=x^{2}\left(x-1\right)-\frac{1}{3}\left(2-x\right)$$

Combine -x^{2} and \frac{2}{5}x^{2} to get -\frac{3}{5}x^{2}.

$$x^{3}-\frac{3}{5}x^{2}-\frac{22}{15}x-\frac{1}{27}+2-x\left(\frac{2}{5}x+3\right)=x^{2}\left(x-1\right)-\frac{1}{3}\left(2-x\right)$$

Add -\frac{1}{27} and 2 to get \frac{53}{27}.

$$x^{3}-\frac{3}{5}x^{2}-\frac{22}{15}x+\frac{53}{27}-x\left(\frac{2}{5}x+3\right)=x^{2}\left(x-1\right)-\frac{1}{3}\left(2-x\right)$$

Use the distributive property to multiply x by \frac{2}{5}x+3.

$$x^{3}-\frac{3}{5}x^{2}-\frac{22}{15}x+\frac{53}{27}-\left(\frac{2}{5}x^{2}+3x\right)=x^{2}\left(x-1\right)-\frac{1}{3}\left(2-x\right)$$

To find the opposite of \frac{2}{5}x^{2}+3x, find the opposite of each term.

$$x^{3}-\frac{3}{5}x^{2}-\frac{22}{15}x+\frac{53}{27}-\frac{2}{5}x^{2}-3x=x^{2}\left(x-1\right)-\frac{1}{3}\left(2-x\right)$$

Combine -\frac{3}{5}x^{2} and -\frac{2}{5}x^{2} to get -x^{2}.

$$x^{3}-x^{2}-\frac{22}{15}x+\frac{53}{27}-3x=x^{2}\left(x-1\right)-\frac{1}{3}\left(2-x\right)$$

Combine -\frac{22}{15}x and -3x to get -\frac{67}{15}x.

$$x^{3}-x^{2}-\frac{67}{15}x+\frac{53}{27}=x^{2}\left(x-1\right)-\frac{1}{3}\left(2-x\right)$$

Use the distributive property to multiply x^{2} by x-1.

$$x^{3}-x^{2}-\frac{67}{15}x+\frac{53}{27}=x^{3}-x^{2}-\frac{1}{3}\left(2-x\right)$$

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

$$x^{3}-x^{2}-\frac{67}{15}x+\frac{53}{27}=x^{3}-x^{2}-\frac{2}{3}+\frac{1}{3}x$$

Subtract x^{3} from both sides.

$$x^{3}-x^{2}-\frac{67}{15}x+\frac{53}{27}-x^{3}=-x^{2}-\frac{2}{3}+\frac{1}{3}x$$

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

$$-x^{2}-\frac{67}{15}x+\frac{53}{27}=-x^{2}-\frac{2}{3}+\frac{1}{3}x$$

Add x^{2} to both sides.

$$-x^{2}-\frac{67}{15}x+\frac{53}{27}+x^{2}=-\frac{2}{3}+\frac{1}{3}x$$

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

$$-\frac{67}{15}x+\frac{53}{27}=-\frac{2}{3}+\frac{1}{3}x$$

Subtract \frac{1}{3}x from both sides.

$$-\frac{67}{15}x+\frac{53}{27}-\frac{1}{3}x=-\frac{2}{3}$$

Combine -\frac{67}{15}x and -\frac{1}{3}x to get -\frac{24}{5}x.

$$-\frac{24}{5}x+\frac{53}{27}=-\frac{2}{3}$$

Subtract \frac{53}{27} from both sides.

$$-\frac{24}{5}x=-\frac{2}{3}-\frac{53}{27}$$

Subtract \frac{53}{27} from -\frac{2}{3} to get -\frac{71}{27}.

$$-\frac{24}{5}x=-\frac{71}{27}$$

Multiply both sides by -\frac{5}{24}, the reciprocal of -\frac{24}{5}.

$$x=-\frac{71}{27}\left(-\frac{5}{24}\right)$$

Multiply -\frac{71}{27} and -\frac{5}{24} to get \frac{355}{648}.

$$x=\frac{355}{648}$$