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

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

Evaluate

$$t^{2}+\frac{3\sqrt{2}t}{2}+\frac{5}{4}$$

Short Solution Steps

Express \frac{\sqrt{2}}{2}t as a single fraction.

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

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

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

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

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

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

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

Express \frac{\sqrt{2}}{2}t as a single fraction.

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

Since \frac{1}{2} and \frac{\sqrt{2}t}{2} have the same denominator, add them by adding their numerators.

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

To raise \frac{1+\sqrt{2}t}{2} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

Do the multiplications in \left(\sqrt{2}t+2\right)^{2}+\left(1+\sqrt{2}t\right)^{2}.

$$\frac{2t^{2}+4\sqrt{2}t+4+1+2\sqrt{2}t+2t^{2}}{2^{2}}$$

Combine like terms in 2t^{2}+4\sqrt{2}t+4+1+2\sqrt{2}t+2t^{2}.

$$\frac{4t^{2}+6\sqrt{2}t+5}{2^{2}}$$

Expand 2^{2}.

$$\frac{4t^{2}+6\sqrt{2}t+5}{4}$$

Expand

$$t^{2}+\frac{3\sqrt{2}t}{2}+\frac{5}{4}$$

Short Solution Steps

Express \frac{\sqrt{2}}{2}t as a single fraction.

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

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

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

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

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

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

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

Express \frac{\sqrt{2}}{2}t as a single fraction.

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

Since \frac{1}{2} and \frac{\sqrt{2}t}{2} have the same denominator, add them by adding their numerators.

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

To raise \frac{1+\sqrt{2}t}{2} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

Do the multiplications in \left(\sqrt{2}t+2\right)^{2}+\left(1+\sqrt{2}t\right)^{2}.

$$\frac{2t^{2}+4\sqrt{2}t+4+1+2\sqrt{2}t+2t^{2}}{2^{2}}$$

Combine like terms in 2t^{2}+4\sqrt{2}t+4+1+2\sqrt{2}t+2t^{2}.

$$\frac{4t^{2}+6\sqrt{2}t+5}{2^{2}}$$

Expand 2^{2}.

$$\frac{4t^{2}+6\sqrt{2}t+5}{4}$$