Solve frac{9+2sqrt{2}}{sqrt{2}-2}

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

Evaluate

$$-\frac{13\sqrt{2}}{2}-11\approx -20.192388155$$

Solution Steps

Rationalize the denominator of \frac{9+2\sqrt{2}}{\sqrt{2}-2} by multiplying numerator and denominator by \sqrt{2}+2.

$$\frac{\left(9+2\sqrt{2}\right)\left(\sqrt{2}+2\right)}{\left(\sqrt{2}-2\right)\left(\sqrt{2}+2\right)}$$

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

$$\frac{\left(9+2\sqrt{2}\right)\left(\sqrt{2}+2\right)}{\left(\sqrt{2}\right)^{2}-2^{2}}$$

Square \sqrt{2}. Square 2.

$$\frac{\left(9+2\sqrt{2}\right)\left(\sqrt{2}+2\right)}{2-4}$$

Subtract 4 from 2 to get -2.

$$\frac{\left(9+2\sqrt{2}\right)\left(\sqrt{2}+2\right)}{-2}$$

Apply the distributive property by multiplying each term of 9+2\sqrt{2} by each term of \sqrt{2}+2.

$$\frac{9\sqrt{2}+18+2\left(\sqrt{2}\right)^{2}+4\sqrt{2}}{-2}$$

The square of \sqrt{2} is 2.

$$\frac{9\sqrt{2}+18+2\times 2+4\sqrt{2}}{-2}$$

Multiply 2 and 2 to get 4.

$$\frac{9\sqrt{2}+18+4+4\sqrt{2}}{-2}$$

Add 18 and 4 to get 22.

$$\frac{9\sqrt{2}+22+4\sqrt{2}}{-2}$$

Combine 9\sqrt{2} and 4\sqrt{2} to get 13\sqrt{2}.

$$\frac{13\sqrt{2}+22}{-2}$$

Multiply both numerator and denominator by -1.

$$\frac{-13\sqrt{2}-22}{2}$$