Solve 2{left(3/2-1/3xright)}^2=left(3/2-1/3xright)left(-frac{1}{3}left(x-1right)+frac{1}{3}left(1-xright)right)

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

Solve for x

$$x = \frac{9}{2} = 4\frac{1}{2} = 4.5$$

Steps for Solving Linear Equation

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

$$2\left(\frac{9}{4}-x+\frac{1}{9}x^{2}\right)=\left(\frac{3}{2}-\frac{1}{3}x\right)\left(-\frac{1}{3}\left(x-1\right)+\frac{1}{3}\left(1-x\right)\right)$$

Use the distributive property to multiply 2 by \frac{9}{4}-x+\frac{1}{9}x^{2}.

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

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

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

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

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

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

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

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

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

Use the distributive property to multiply \frac{3}{2}-\frac{1}{3}x by -\frac{2}{3}x+\frac{2}{3} and combine like terms.

$$\frac{9}{2}-2x+\frac{2}{9}x^{2}=-\frac{11}{9}x+1+\frac{2}{9}x^{2}$$

Add \frac{11}{9}x to both sides.

$$\frac{9}{2}-2x+\frac{2}{9}x^{2}+\frac{11}{9}x=1+\frac{2}{9}x^{2}$$

Combine -2x and \frac{11}{9}x to get -\frac{7}{9}x.

$$\frac{9}{2}-\frac{7}{9}x+\frac{2}{9}x^{2}=1+\frac{2}{9}x^{2}$$

Subtract \frac{2}{9}x^{2} from both sides.

$$\frac{9}{2}-\frac{7}{9}x+\frac{2}{9}x^{2}-\frac{2}{9}x^{2}=1$$

Combine \frac{2}{9}x^{2} and -\frac{2}{9}x^{2} to get 0.

$$\frac{9}{2}-\frac{7}{9}x=1$$

Subtract \frac{9}{2} from both sides.

$$-\frac{7}{9}x=1-\frac{9}{2}$$

Subtract \frac{9}{2} from 1 to get -\frac{7}{2}.

$$-\frac{7}{9}x=-\frac{7}{2}$$

Multiply both sides by -\frac{9}{7}, the reciprocal of -\frac{7}{9}.

$$x=-\frac{7}{2}\left(-\frac{9}{7}\right)$$

Multiply -\frac{7}{2} and -\frac{9}{7} to get \frac{9}{2}.

$$x=\frac{9}{2}$$