Solve -1/2(1+frac{sqrt{2}}{2})^2+2(1+frac{sqrt{2}}{2})

$$- \frac { 1 } { 2 } ( 1 + \frac { \sqrt { 2 } } { 2 } ) ^ { 2 } + 2 ( 1 + \frac { \sqrt { 2 } } { 2 } )$$

Evaluate

$$\frac{\sqrt{2}}{2}+\frac{5}{4}\approx 1.957106781$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{2}{2}.

$$-\frac{1}{2}\left(\frac{2}{2}+\frac{\sqrt{2}}{2}\right)^{2}+2\left(1+\frac{\sqrt{2}}{2}\right)$$

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

$$-\frac{1}{2}\times \left(\frac{2+\sqrt{2}}{2}\right)^{2}+2\left(1+\frac{\sqrt{2}}{2}\right)$$

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

$$-\frac{1}{2}\times \frac{\left(2+\sqrt{2}\right)^{2}}{2^{2}}+2\left(1+\frac{\sqrt{2}}{2}\right)$$

Multiply -\frac{1}{2} times \frac{\left(2+\sqrt{2}\right)^{2}}{2^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-\left(2+\sqrt{2}\right)^{2}}{2\times 2^{2}}+2\left(1+\frac{\sqrt{2}}{2}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{2}{2}.

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

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

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

Cancel out 2 and 2.

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

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

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

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

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

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\frac{-\left(2+\sqrt{2}\right)^{2}}{2^{3}}+2+\sqrt{2}$$

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

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

The square of \sqrt{2} is 2.

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

Add 4 and 2 to get 6.

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

Calculate 2 to the power of 3 and get 8.

$$\frac{-\left(6+4\sqrt{2}\right)}{8}+2+\sqrt{2}$$

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

$$\frac{-\left(6+4\sqrt{2}\right)}{8}+\frac{8\left(2+\sqrt{2}\right)}{8}$$

Since \frac{-\left(6+4\sqrt{2}\right)}{8} and \frac{8\left(2+\sqrt{2}\right)}{8} have the same denominator, add them by adding their numerators.

$$\frac{-\left(6+4\sqrt{2}\right)+8\left(2+\sqrt{2}\right)}{8}$$

Do the multiplications in -\left(6+4\sqrt{2}\right)+8\left(2+\sqrt{2}\right).

$$\frac{-6-4\sqrt{2}+16+8\sqrt{2}}{8}$$

Do the calculations in -6-4\sqrt{2}+16+8\sqrt{2}.

$$\frac{10+4\sqrt{2}}{8}$$

Factor

$$\frac{2 \sqrt{2} + 5}{4} = 1.9571067811865475$$