Solve 1/2x(3-x)+3(x+1)(x-1)-x(x-1)^2+(x-1)^3-1/2(2x-8)

$$\frac { 1 } { 2 } x ( 3 - x ) + 3 ( x + 1 ) ( x - 1 ) - x ( x - 1 ) ^ { 2 } + ( x - 1 ) ^ { 3 } - \frac { 1 } { 2 } ( 2 x - 8 )$$

Evaluate

$$\frac{x\left(3x+5\right)}{2}$$

Solution Steps

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

$$\frac{3}{2}x-\frac{1}{2}x^{2}+3\left(x+1\right)\left(x-1\right)-x\left(x-1\right)^{2}+\left(x-1\right)^{3}-\frac{1}{2}\left(2x-8\right)$$

Use the distributive property to multiply 3 by x+1.

$$\frac{3}{2}x-\frac{1}{2}x^{2}+\left(3x+3\right)\left(x-1\right)-x\left(x-1\right)^{2}+\left(x-1\right)^{3}-\frac{1}{2}\left(2x-8\right)$$

Use the distributive property to multiply 3x+3 by x-1 and combine like terms.

$$\frac{3}{2}x-\frac{1}{2}x^{2}+3x^{2}-3-x\left(x-1\right)^{2}+\left(x-1\right)^{3}-\frac{1}{2}\left(2x-8\right)$$

Combine -\frac{1}{2}x^{2} and 3x^{2} to get \frac{5}{2}x^{2}.

$$\frac{3}{2}x+\frac{5}{2}x^{2}-3-x\left(x-1\right)^{2}+\left(x-1\right)^{3}-\frac{1}{2}\left(2x-8\right)$$

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

$$\frac{3}{2}x+\frac{5}{2}x^{2}-3-x\left(x^{2}-2x+1\right)+\left(x-1\right)^{3}-\frac{1}{2}\left(2x-8\right)$$

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

$$\frac{3}{2}x+\frac{5}{2}x^{2}-3-\left(x^{3}-2x^{2}+x\right)+\left(x-1\right)^{3}-\frac{1}{2}\left(2x-8\right)$$

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

$$\frac{3}{2}x+\frac{5}{2}x^{2}-3-x^{3}+2x^{2}-x+\left(x-1\right)^{3}-\frac{1}{2}\left(2x-8\right)$$

Combine \frac{5}{2}x^{2} and 2x^{2} to get \frac{9}{2}x^{2}.

$$\frac{3}{2}x+\frac{9}{2}x^{2}-3-x^{3}-x+\left(x-1\right)^{3}-\frac{1}{2}\left(2x-8\right)$$

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

$$\frac{1}{2}x+\frac{9}{2}x^{2}-3-x^{3}+\left(x-1\right)^{3}-\frac{1}{2}\left(2x-8\right)$$

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

$$\frac{1}{2}x+\frac{9}{2}x^{2}-3-x^{3}+x^{3}-3x^{2}+3x-1-\frac{1}{2}\left(2x-8\right)$$

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

$$\frac{1}{2}x+\frac{9}{2}x^{2}-3-3x^{2}+3x-1-\frac{1}{2}\left(2x-8\right)$$

Combine \frac{9}{2}x^{2} and -3x^{2} to get \frac{3}{2}x^{2}.

$$\frac{1}{2}x+\frac{3}{2}x^{2}-3+3x-1-\frac{1}{2}\left(2x-8\right)$$

Combine \frac{1}{2}x and 3x to get \frac{7}{2}x.

$$\frac{7}{2}x+\frac{3}{2}x^{2}-3-1-\frac{1}{2}\left(2x-8\right)$$

Subtract 1 from -3 to get -4.

$$\frac{7}{2}x+\frac{3}{2}x^{2}-4-\frac{1}{2}\left(2x-8\right)$$

Use the distributive property to multiply -\frac{1}{2} by 2x-8.

$$\frac{7}{2}x+\frac{3}{2}x^{2}-4-x+4$$

Combine \frac{7}{2}x and -x to get \frac{5}{2}x.

$$\frac{5}{2}x+\frac{3}{2}x^{2}-4+4$$

Add -4 and 4 to get 0.

$$\frac{5}{2}x+\frac{3}{2}x^{2}$$

Expand

$$\frac{3x^{2}+5x}{2}$$

Solution Steps

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

$$\frac{3}{2}x-\frac{1}{2}x^{2}+3\left(x+1\right)\left(x-1\right)-x\left(x-1\right)^{2}+\left(x-1\right)^{3}-\frac{1}{2}\left(2x-8\right)$$

Use the distributive property to multiply 3 by x+1.

$$\frac{3}{2}x-\frac{1}{2}x^{2}+\left(3x+3\right)\left(x-1\right)-x\left(x-1\right)^{2}+\left(x-1\right)^{3}-\frac{1}{2}\left(2x-8\right)$$

Use the distributive property to multiply 3x+3 by x-1 and combine like terms.

$$\frac{3}{2}x-\frac{1}{2}x^{2}+3x^{2}-3-x\left(x-1\right)^{2}+\left(x-1\right)^{3}-\frac{1}{2}\left(2x-8\right)$$

Combine -\frac{1}{2}x^{2} and 3x^{2} to get \frac{5}{2}x^{2}.

$$\frac{3}{2}x+\frac{5}{2}x^{2}-3-x\left(x-1\right)^{2}+\left(x-1\right)^{3}-\frac{1}{2}\left(2x-8\right)$$

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

$$\frac{3}{2}x+\frac{5}{2}x^{2}-3-x\left(x^{2}-2x+1\right)+\left(x-1\right)^{3}-\frac{1}{2}\left(2x-8\right)$$

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

$$\frac{3}{2}x+\frac{5}{2}x^{2}-3-\left(x^{3}-2x^{2}+x\right)+\left(x-1\right)^{3}-\frac{1}{2}\left(2x-8\right)$$

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

$$\frac{3}{2}x+\frac{5}{2}x^{2}-3-x^{3}+2x^{2}-x+\left(x-1\right)^{3}-\frac{1}{2}\left(2x-8\right)$$

Combine \frac{5}{2}x^{2} and 2x^{2} to get \frac{9}{2}x^{2}.

$$\frac{3}{2}x+\frac{9}{2}x^{2}-3-x^{3}-x+\left(x-1\right)^{3}-\frac{1}{2}\left(2x-8\right)$$

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

$$\frac{1}{2}x+\frac{9}{2}x^{2}-3-x^{3}+\left(x-1\right)^{3}-\frac{1}{2}\left(2x-8\right)$$

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

$$\frac{1}{2}x+\frac{9}{2}x^{2}-3-x^{3}+x^{3}-3x^{2}+3x-1-\frac{1}{2}\left(2x-8\right)$$

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

$$\frac{1}{2}x+\frac{9}{2}x^{2}-3-3x^{2}+3x-1-\frac{1}{2}\left(2x-8\right)$$

Combine \frac{9}{2}x^{2} and -3x^{2} to get \frac{3}{2}x^{2}.

$$\frac{1}{2}x+\frac{3}{2}x^{2}-3+3x-1-\frac{1}{2}\left(2x-8\right)$$

Combine \frac{1}{2}x and 3x to get \frac{7}{2}x.

$$\frac{7}{2}x+\frac{3}{2}x^{2}-3-1-\frac{1}{2}\left(2x-8\right)$$

Subtract 1 from -3 to get -4.

$$\frac{7}{2}x+\frac{3}{2}x^{2}-4-\frac{1}{2}\left(2x-8\right)$$

Use the distributive property to multiply -\frac{1}{2} by 2x-8.

$$\frac{7}{2}x+\frac{3}{2}x^{2}-4-x+4$$

Combine \frac{7}{2}x and -x to get \frac{5}{2}x.

$$\frac{5}{2}x+\frac{3}{2}x^{2}-4+4$$

Add -4 and 4 to get 0.

$$\frac{5}{2}x+\frac{3}{2}x^{2}$$