Solve (x-1/sqrt{2})(x+1/sqrt{2})

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

Evaluate

$$x^{2}-\frac{1}{2}$$

Short Solution Steps

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

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

The square of \sqrt{2} is 2.

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

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

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

Since \frac{2x}{2} and \frac{\sqrt{2}}{2} have the same denominator, subtract them by subtracting their numerators.

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

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

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

The square of \sqrt{2} is 2.

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

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

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

Since \frac{2x}{2} and \frac{\sqrt{2}}{2} have the same denominator, add them by adding their numerators.

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

Multiply \frac{2x-\sqrt{2}}{2} times \frac{2x+\sqrt{2}}{2} by multiplying numerator times numerator and denominator times denominator.

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

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

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

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

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{2} is 2.

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

Multiply 2 and 2 to get 4.

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

Differentiate w.r.t. x

$$2x$$