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

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

Evaluate

$$-\frac{9\sqrt{2}}{2}+5\approx -1.363961031$$

Solution Steps

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}. Square 2.

$$4-\left(\sqrt{5}\right)^{2}+\left(2-\sqrt{2}\right)^{2}-\frac{1}{\sqrt{2}}$$

The square of \sqrt{5} is 5.

$$4-5+\left(2-\sqrt{2}\right)^{2}-\frac{1}{\sqrt{2}}$$

Subtract 5 from 4 to get -1.

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

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

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

The square of \sqrt{2} is 2.

$$-1+4-4\sqrt{2}+2-\frac{1}{\sqrt{2}}$$

Add 4 and 2 to get 6.

$$-1+6-4\sqrt{2}-\frac{1}{\sqrt{2}}$$

Add -1 and 6 to get 5.

$$5-4\sqrt{2}-\frac{1}{\sqrt{2}}$$

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

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

The square of \sqrt{2} is 2.

$$5-4\sqrt{2}-\frac{\sqrt{2}}{2}$$

Combine -4\sqrt{2} and -\frac{\sqrt{2}}{2} to get -\frac{9}{2}\sqrt{2}.

$$5-\frac{9}{2}\sqrt{2}$$