Solve frac{-3+2sqrt{2}}{1-sqrt{2}}

$$\frac { - 3 + 2 \sqrt { 2 } } { 1 - \sqrt { 2 } }$$

Evaluate

$$\sqrt{2}-1\approx 0.414213562$$

Solution Steps

Rationalize the denominator of \frac{-3+2\sqrt{2}}{1-\sqrt{2}} by multiplying numerator and denominator by 1+\sqrt{2}.

$$\frac{\left(-3+2\sqrt{2}\right)\left(1+\sqrt{2}\right)}{\left(1-\sqrt{2}\right)\left(1+\sqrt{2}\right)}$$

Consider \left(1-\sqrt{2}\right)\left(1+\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}.

$$\frac{\left(-3+2\sqrt{2}\right)\left(1+\sqrt{2}\right)}{1^{2}-\left(\sqrt{2}\right)^{2}}$$

Square 1. Square \sqrt{2}.

$$\frac{\left(-3+2\sqrt{2}\right)\left(1+\sqrt{2}\right)}{1-2}$$

Subtract 2 from 1 to get -1.

$$\frac{\left(-3+2\sqrt{2}\right)\left(1+\sqrt{2}\right)}{-1}$$

Anything divided by -1 gives its opposite.

$$-\left(-3+2\sqrt{2}\right)\left(1+\sqrt{2}\right)$$

Apply the distributive property by multiplying each term of -3+2\sqrt{2} by each term of 1+\sqrt{2}.

$$-\left(-3-3\sqrt{2}+2\sqrt{2}+2\left(\sqrt{2}\right)^{2}\right)$$

Combine -3\sqrt{2} and 2\sqrt{2} to get -\sqrt{2}.

$$-\left(-3-\sqrt{2}+2\left(\sqrt{2}\right)^{2}\right)$$

The square of \sqrt{2} is 2.

$$-\left(-3-\sqrt{2}+2\times 2\right)$$

Multiply 2 and 2 to get 4.

$$-\left(-3-\sqrt{2}+4\right)$$

Add -3 and 4 to get 1.

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

To find the opposite of 1-\sqrt{2}, find the opposite of each term.

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

The opposite of -\sqrt{2} is \sqrt{2}.

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