Solve (frac{sqrt{17}+1}{4})^2+(frac{1-sqrt{17}}{4})^2

$$( \frac { \sqrt { 17 } + 1 } { 4 } ) ^ { 2 } + ( \frac { 1 - \sqrt { 17 } } { 4 } ) ^ { 2 }$$

Evaluate

$$\frac{9}{4}=2.25$$

Short Solution Steps

To raise \frac{\sqrt{17}+1}{4} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(\sqrt{17}+1\right)^{2}}{4^{2}}+\left(\frac{1-\sqrt{17}}{4}\right)^{2}$$

To raise \frac{1-\sqrt{17}}{4} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(\sqrt{17}+1\right)^{2}}{4^{2}}+\frac{\left(1-\sqrt{17}\right)^{2}}{4^{2}}$$

Since \frac{\left(\sqrt{17}+1\right)^{2}}{4^{2}} and \frac{\left(1-\sqrt{17}\right)^{2}}{4^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(\sqrt{17}+1\right)^{2}+\left(1-\sqrt{17}\right)^{2}}{4^{2}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{17}+1\right)^{2}.

$$\frac{\left(\sqrt{17}\right)^{2}+2\sqrt{17}+1}{4^{2}}+\frac{\left(1-\sqrt{17}\right)^{2}}{4^{2}}$$

The square of \sqrt{17} is 17.

$$\frac{17+2\sqrt{17}+1}{4^{2}}+\frac{\left(1-\sqrt{17}\right)^{2}}{4^{2}}$$

Add 17 and 1 to get 18.

$$\frac{18+2\sqrt{17}}{4^{2}}+\frac{\left(1-\sqrt{17}\right)^{2}}{4^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{18+2\sqrt{17}}{16}+\frac{\left(1-\sqrt{17}\right)^{2}}{4^{2}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-\sqrt{17}\right)^{2}.

$$\frac{18+2\sqrt{17}}{16}+\frac{1-2\sqrt{17}+\left(\sqrt{17}\right)^{2}}{4^{2}}$$

The square of \sqrt{17} is 17.

$$\frac{18+2\sqrt{17}}{16}+\frac{1-2\sqrt{17}+17}{4^{2}}$$

Add 1 and 17 to get 18.

$$\frac{18+2\sqrt{17}}{16}+\frac{18-2\sqrt{17}}{4^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{18+2\sqrt{17}}{16}+\frac{18-2\sqrt{17}}{16}$$

Since \frac{18+2\sqrt{17}}{16} and \frac{18-2\sqrt{17}}{16} have the same denominator, add them by adding their numerators.

$$\frac{18+2\sqrt{17}+18-2\sqrt{17}}{16}$$

Do the calculations in 18+2\sqrt{17}+18-2\sqrt{17}.

$$\frac{36}{16}$$

Factor

$$\frac{3 ^ {2}}{2 ^ {2}} = 2\frac{1}{4} = 2.25$$