Solve 2/x-sqrt{2}-x-sqrt{2}

$$\frac{ 2 }{ x- \sqrt{ 2 } } -x- \sqrt{ 2 }$$

Evaluate

$$-\frac{x^{2}-4}{x-\sqrt{2}}$$

Short Solution Steps

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

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

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

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

The square of \sqrt{2} is 2.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply -x-\sqrt{2} times \frac{x^{2}-2}{x^{2}-2}.

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

Since \frac{2\left(x+\sqrt{2}\right)}{x^{2}-2} and \frac{\left(-x-\sqrt{2}\right)\left(x^{2}-2\right)}{x^{2}-2} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 2\left(x+\sqrt{2}\right)+\left(-x-\sqrt{2}\right)\left(x^{2}-2\right).

$$\frac{2x+2\sqrt{2}-x^{3}+2x-\sqrt{2}x^{2}+2\sqrt{2}}{x^{2}-2}$$

Combine like terms in 2x+2\sqrt{2}-x^{3}+2x-\sqrt{2}x^{2}+2\sqrt{2}.

$$\frac{4x-\sqrt{2}x^{2}+4\sqrt{2}-x^{3}}{x^{2}-2}$$

Differentiate w.r.t. x

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