Solve (frac{x^{2}-2x+3}{2}-x)^2+(x^2-2x-1/2-x^2+2x+4)^2

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

Evaluate

$$\frac{x^{4}}{2}-3x^{3}+3x^{2}+x+\frac{29}{2}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{2}{2}.

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

Since \frac{x^{2}-2x+3}{2} and \frac{2x}{2} have the same denominator, subtract them by subtracting their numerators.

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

Combine like terms in x^{2}-2x+3-2x.

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

To raise \frac{x^{2}-4x+3}{2} to a power, raise both numerator and denominator to the power and then divide.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply -x^{2}+2x+4 times \frac{2}{2}.

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

Since \frac{x^{2}-2x-1}{2} and \frac{2\left(-x^{2}+2x+4\right)}{2} have the same denominator, add them by adding their numerators.

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

Do the multiplications in x^{2}-2x-1+2\left(-x^{2}+2x+4\right).

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

Combine like terms in x^{2}-2x-1-2x^{2}+4x+8.

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

To raise \frac{-x^{2}+2x+7}{2} to a power, raise both numerator and denominator to the power and then divide.

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

Since \frac{\left(x^{2}-4x+3\right)^{2}}{2^{2}} and \frac{\left(-x^{2}+2x+7\right)^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(x^{2}-4x+3\right)^{2}+\left(-x^{2}+2x+7\right)^{2}.

$$\frac{x^{4}-4x^{3}+3x^{2}-4x^{3}+16x^{2}-12x+3x^{2}-12x+9+x^{4}-2x^{3}-7x^{2}-2x^{3}+4x^{2}+14x-7x^{2}+14x+49}{2^{2}}$$

Combine like terms in x^{4}-4x^{3}+3x^{2}-4x^{3}+16x^{2}-12x+3x^{2}-12x+9+x^{4}-2x^{3}-7x^{2}-2x^{3}+4x^{2}+14x-7x^{2}+14x+49.

$$\frac{2x^{4}-12x^{3}+12x^{2}+4x+58}{2^{2}}$$

Factor the expressions that are not already factored in \frac{2x^{4}-12x^{3}+12x^{2}+4x+58}{2^{2}}.

$$\frac{2\left(x^{4}-6x^{3}+6x^{2}+2x+29\right)}{2^{2}}$$

Cancel out 2 in both numerator and denominator.

$$\frac{x^{4}-6x^{3}+6x^{2}+2x+29}{2}$$

Expand

$$\frac{x^{4}}{2}-3x^{3}+3x^{2}+x+\frac{29}{2}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{2}{2}.

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

Since \frac{x^{2}-2x+3}{2} and \frac{2x}{2} have the same denominator, subtract them by subtracting their numerators.

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

Combine like terms in x^{2}-2x+3-2x.

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

To raise \frac{x^{2}-4x+3}{2} to a power, raise both numerator and denominator to the power and then divide.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply -x^{2}+2x+4 times \frac{2}{2}.

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

Since \frac{x^{2}-2x-1}{2} and \frac{2\left(-x^{2}+2x+4\right)}{2} have the same denominator, add them by adding their numerators.

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

Do the multiplications in x^{2}-2x-1+2\left(-x^{2}+2x+4\right).

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

Combine like terms in x^{2}-2x-1-2x^{2}+4x+8.

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

To raise \frac{-x^{2}+2x+7}{2} to a power, raise both numerator and denominator to the power and then divide.

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

Since \frac{\left(x^{2}-4x+3\right)^{2}}{2^{2}} and \frac{\left(-x^{2}+2x+7\right)^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(x^{2}-4x+3\right)^{2}+\left(-x^{2}+2x+7\right)^{2}.

$$\frac{x^{4}-4x^{3}+3x^{2}-4x^{3}+16x^{2}-12x+3x^{2}-12x+9+x^{4}-2x^{3}-7x^{2}-2x^{3}+4x^{2}+14x-7x^{2}+14x+49}{2^{2}}$$

Combine like terms in x^{4}-4x^{3}+3x^{2}-4x^{3}+16x^{2}-12x+3x^{2}-12x+9+x^{4}-2x^{3}-7x^{2}-2x^{3}+4x^{2}+14x-7x^{2}+14x+49.

$$\frac{2x^{4}-12x^{3}+12x^{2}+4x+58}{2^{2}}$$

Factor the expressions that are not already factored in \frac{2x^{4}-12x^{3}+12x^{2}+4x+58}{2^{2}}.

$$\frac{2\left(x^{4}-6x^{3}+6x^{2}+2x+29\right)}{2^{2}}$$

Cancel out 2 in both numerator and denominator.

$$\frac{x^{4}-6x^{3}+6x^{2}+2x+29}{2}$$