Solve x^2+{left(-x/2+2right)}^2-2x-4(-x/2+2)

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

Evaluate

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

Short Solution Steps

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

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

Since -\frac{x}{2} and \frac{2\times 2}{2} have the same denominator, add them by adding their numerators.

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

Do the multiplications in -x+2\times 2.

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

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

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

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

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

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

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

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

$$\frac{4x^{2}-8x+x^{2}-8x+16}{2^{2}}-4\left(-\frac{x}{2}+2\right)$$

Combine like terms in 4x^{2}-8x+x^{2}-8x+16.

$$\frac{5x^{2}-16x+16}{2^{2}}-4\left(-\frac{x}{2}+2\right)$$

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

$$\frac{5x^{2}-16x+16}{2^{2}}-4\left(-\frac{x}{2}+\frac{2\times 2}{2}\right)$$

Since -\frac{x}{2} and \frac{2\times 2}{2} have the same denominator, add them by adding their numerators.

$$\frac{5x^{2}-16x+16}{2^{2}}-4\times \frac{-x+2\times 2}{2}$$

Do the multiplications in -x+2\times 2.

$$\frac{5x^{2}-16x+16}{2^{2}}-4\times \frac{-x+4}{2}$$

Cancel out 2, the greatest common factor in 4 and 2.

$$\frac{5x^{2}-16x+16}{2^{2}}-2\left(-x+4\right)$$

Calculate 2 to the power of 2 and get 4.

$$\frac{5x^{2}-16x+16}{4}-2\left(-x+4\right)$$

Use the distributive property to multiply -2 by -x+4.

$$\frac{5x^{2}-16x+16}{4}+2x-8$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2x-8 times \frac{4}{4}.

$$\frac{5x^{2}-16x+16}{4}+\frac{4\left(2x-8\right)}{4}$$

Since \frac{5x^{2}-16x+16}{4} and \frac{4\left(2x-8\right)}{4} have the same denominator, add them by adding their numerators.

$$\frac{5x^{2}-16x+16+4\left(2x-8\right)}{4}$$

Do the multiplications in 5x^{2}-16x+16+4\left(2x-8\right).

$$\frac{5x^{2}-16x+16+8x-32}{4}$$

Combine like terms in 5x^{2}-16x+16+8x-32.

$$\frac{5x^{2}-8x-16}{4}$$

Expand

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

Short Solution Steps

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

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

Since -\frac{x}{2} and \frac{2\times 2}{2} have the same denominator, add them by adding their numerators.

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

Do the multiplications in -x+2\times 2.

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

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

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

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

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

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

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

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

$$\frac{4x^{2}-8x+x^{2}-8x+16}{2^{2}}-4\left(-\frac{x}{2}+2\right)$$

Combine like terms in 4x^{2}-8x+x^{2}-8x+16.

$$\frac{5x^{2}-16x+16}{2^{2}}-4\left(-\frac{x}{2}+2\right)$$

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

$$\frac{5x^{2}-16x+16}{2^{2}}-4\left(-\frac{x}{2}+\frac{2\times 2}{2}\right)$$

Since -\frac{x}{2} and \frac{2\times 2}{2} have the same denominator, add them by adding their numerators.

$$\frac{5x^{2}-16x+16}{2^{2}}-4\times \frac{-x+2\times 2}{2}$$

Do the multiplications in -x+2\times 2.

$$\frac{5x^{2}-16x+16}{2^{2}}-4\times \frac{-x+4}{2}$$

Cancel out 2, the greatest common factor in 4 and 2.

$$\frac{5x^{2}-16x+16}{2^{2}}-2\left(-x+4\right)$$

Calculate 2 to the power of 2 and get 4.

$$\frac{5x^{2}-16x+16}{4}-2\left(-x+4\right)$$

Use the distributive property to multiply -2 by -x+4.

$$\frac{5x^{2}-16x+16}{4}+2x-8$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2x-8 times \frac{4}{4}.

$$\frac{5x^{2}-16x+16}{4}+\frac{4\left(2x-8\right)}{4}$$

Since \frac{5x^{2}-16x+16}{4} and \frac{4\left(2x-8\right)}{4} have the same denominator, add them by adding their numerators.

$$\frac{5x^{2}-16x+16+4\left(2x-8\right)}{4}$$

Do the multiplications in 5x^{2}-16x+16+4\left(2x-8\right).

$$\frac{5x^{2}-16x+16+8x-32}{4}$$

Combine like terms in 5x^{2}-16x+16+8x-32.

$$\frac{5x^{2}-8x-16}{4}$$