Solve {left(frac{1-sqrt{53}}{4}right)}^2+{left(frac{1+sqrt{53}}{4}right)}^2

$${ \left( \frac{ 1- \sqrt{ 53 } }{ 4 } \right) }^{ 2 } + { \left( \frac{ 1+ \sqrt{ 53 } }{ 4 } \right) }^{ 2 }$$

Evaluate

$$\frac{27}{4}=6.75$$

Short Solution Steps

To raise \frac{1-\sqrt{53}}{4} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(1-\sqrt{53}\right)^{2}}{4^{2}}+\left(\frac{1+\sqrt{53}}{4}\right)^{2}$$

To raise \frac{1+\sqrt{53}}{4} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(1-\sqrt{53}\right)^{2}}{4^{2}}+\frac{\left(1+\sqrt{53}\right)^{2}}{4^{2}}$$

Since \frac{\left(1-\sqrt{53}\right)^{2}}{4^{2}} and \frac{\left(1+\sqrt{53}\right)^{2}}{4^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(1-\sqrt{53}\right)^{2}+\left(1+\sqrt{53}\right)^{2}}{4^{2}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-\sqrt{53}\right)^{2}.

$$\frac{1-2\sqrt{53}+\left(\sqrt{53}\right)^{2}}{4^{2}}+\frac{\left(1+\sqrt{53}\right)^{2}}{4^{2}}$$

The square of \sqrt{53} is 53.

$$\frac{1-2\sqrt{53}+53}{4^{2}}+\frac{\left(1+\sqrt{53}\right)^{2}}{4^{2}}$$

Add 1 and 53 to get 54.

$$\frac{54-2\sqrt{53}}{4^{2}}+\frac{\left(1+\sqrt{53}\right)^{2}}{4^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{54-2\sqrt{53}}{16}+\frac{\left(1+\sqrt{53}\right)^{2}}{4^{2}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+\sqrt{53}\right)^{2}.

$$\frac{54-2\sqrt{53}}{16}+\frac{1+2\sqrt{53}+\left(\sqrt{53}\right)^{2}}{4^{2}}$$

The square of \sqrt{53} is 53.

$$\frac{54-2\sqrt{53}}{16}+\frac{1+2\sqrt{53}+53}{4^{2}}$$

Add 1 and 53 to get 54.

$$\frac{54-2\sqrt{53}}{16}+\frac{54+2\sqrt{53}}{4^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{54-2\sqrt{53}}{16}+\frac{54+2\sqrt{53}}{16}$$

Since \frac{54-2\sqrt{53}}{16} and \frac{54+2\sqrt{53}}{16} have the same denominator, add them by adding their numerators.

$$\frac{54-2\sqrt{53}+54+2\sqrt{53}}{16}$$

Do the calculations in 54-2\sqrt{53}+54+2\sqrt{53}.

$$\frac{108}{16}$$

Factor

$$\frac{3 ^ {3}}{2 ^ {2}} = 6\frac{3}{4} = 6.75$$