Solve (2+sqrt{5}+1/2+sqrt{5})^2

$$( 2 + \sqrt { 5 } + \frac { 1 } { 2 + \sqrt { 5 } } ) ^ { 2 }$$

Evaluate

$$20$$

Solution Steps

Rationalize the denominator of \frac{1}{2+\sqrt{5}} by multiplying numerator and denominator by 2-\sqrt{5}.

$$\left(2+\sqrt{5}+\frac{2-\sqrt{5}}{\left(2+\sqrt{5}\right)\left(2-\sqrt{5}\right)}\right)^{2}$$

Consider \left(2+\sqrt{5}\right)\left(2-\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(2+\sqrt{5}+\frac{2-\sqrt{5}}{2^{2}-\left(\sqrt{5}\right)^{2}}\right)^{2}$$

Square 2. Square \sqrt{5}.

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

Subtract 5 from 4 to get -1.

$$\left(2+\sqrt{5}+\frac{2-\sqrt{5}}{-1}\right)^{2}$$

Anything divided by -1 gives its opposite. To find the opposite of 2-\sqrt{5}, find the opposite of each term.

$$\left(2+\sqrt{5}-2+\sqrt{5}\right)^{2}$$

Subtract 2 from 2 to get 0.

$$\left(\sqrt{5}+\sqrt{5}\right)^{2}$$

Combine \sqrt{5} and \sqrt{5} to get 2\sqrt{5}.

$$\left(2\sqrt{5}\right)^{2}$$

Expand \left(2\sqrt{5}\right)^{2}.

$$2^{2}\left(\sqrt{5}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$4\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{5} is 5.

$$4\times 5$$

Multiply 4 and 5 to get 20.

$$20$$

Factor

$$2^{2}\times 5$$