Solve -9/4+2sqrt{2}

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

Evaluate

$$\frac{9\left(\sqrt{2}-2\right)}{4}\approx -1.318019485$$

Solution Steps

Rationalize the denominator of \frac{-9}{4+2\sqrt{2}} by multiplying numerator and denominator by 4-2\sqrt{2}.

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

Consider \left(4+2\sqrt{2}\right)\left(4-2\sqrt{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}.

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

Calculate 4 to the power of 2 and get 16.

$$\frac{-9\left(4-2\sqrt{2}\right)}{16-\left(2\sqrt{2}\right)^{2}}$$

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

$$\frac{-9\left(4-2\sqrt{2}\right)}{16-2^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{-9\left(4-2\sqrt{2}\right)}{16-4\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{-9\left(4-2\sqrt{2}\right)}{16-4\times 2}$$

Multiply 4 and 2 to get 8.

$$\frac{-9\left(4-2\sqrt{2}\right)}{16-8}$$

Subtract 8 from 16 to get 8.

$$\frac{-9\left(4-2\sqrt{2}\right)}{8}$$

Use the distributive property to multiply -9 by 4-2\sqrt{2}.

$$\frac{-36+18\sqrt{2}}{8}$$