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

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

Evaluate

$$\frac{\left(x-1\right)^{3}-4\left(x-2\right)^{2}}{16}$$

Solution Steps

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

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

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

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

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

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

Use the distributive property to multiply \frac{1}{8} by x^{3}-3x^{2}+3x-1.

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

Combine -\frac{1}{2}x^{2} and -\frac{3}{8}x^{2} to get -\frac{7}{8}x^{2}.

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

Combine 2x and \frac{3}{8}x to get \frac{19}{8}x.

$$\frac{1}{2}\left(-\frac{7}{8}x^{2}+\frac{19}{8}x-2+\frac{1}{8}x^{3}-\frac{1}{8}\right)$$

Subtract \frac{1}{8} from -2 to get -\frac{17}{8}.

$$\frac{1}{2}\left(-\frac{7}{8}x^{2}+\frac{19}{8}x-\frac{17}{8}+\frac{1}{8}x^{3}\right)$$

Use the distributive property to multiply \frac{1}{2} by -\frac{7}{8}x^{2}+\frac{19}{8}x-\frac{17}{8}+\frac{1}{8}x^{3}.

$$-\frac{7}{16}x^{2}+\frac{19}{16}x-\frac{17}{16}+\frac{1}{16}x^{3}$$

Expand

$$\frac{x^{3}-7x^{2}+19x-17}{16}$$

Solution Steps

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

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

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

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

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

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

Use the distributive property to multiply \frac{1}{8} by x^{3}-3x^{2}+3x-1.

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

Combine -\frac{1}{2}x^{2} and -\frac{3}{8}x^{2} to get -\frac{7}{8}x^{2}.

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

Combine 2x and \frac{3}{8}x to get \frac{19}{8}x.

$$\frac{1}{2}\left(-\frac{7}{8}x^{2}+\frac{19}{8}x-2+\frac{1}{8}x^{3}-\frac{1}{8}\right)$$

Subtract \frac{1}{8} from -2 to get -\frac{17}{8}.

$$\frac{1}{2}\left(-\frac{7}{8}x^{2}+\frac{19}{8}x-\frac{17}{8}+\frac{1}{8}x^{3}\right)$$

Use the distributive property to multiply \frac{1}{2} by -\frac{7}{8}x^{2}+\frac{19}{8}x-\frac{17}{8}+\frac{1}{8}x^{3}.

$$-\frac{7}{16}x^{2}+\frac{19}{16}x-\frac{17}{16}+\frac{1}{16}x^{3}$$