Solve (x-frac{-sqrt{2}-1}{2})(x-frac{sqrt{2}-1}{2})

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

Evaluate

$$x^{2}+x-\frac{1}{4}$$

Short Solution Steps

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

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

Since \frac{2x}{2} and \frac{-\sqrt{2}-1}{2} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2x-\left(-\sqrt{2}-1\right).

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

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

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

Since \frac{2x}{2} and \frac{\sqrt{2}-1}{2} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2x-\left(\sqrt{2}-1\right).

$$\frac{2x+\sqrt{2}+1}{2}\times \frac{2x-\sqrt{2}+1}{2}$$

Multiply \frac{2x+\sqrt{2}+1}{2} times \frac{2x-\sqrt{2}+1}{2} by multiplying numerator times numerator and denominator times denominator.

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

Multiply 2 and 2 to get 4.

$$\frac{\left(2x+\sqrt{2}+1\right)\left(2x-\sqrt{2}+1\right)}{4}$$

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

$$\frac{4x^{2}-2\sqrt{2}x+2x+2\sqrt{2}x-\left(\sqrt{2}\right)^{2}+\sqrt{2}+2x-\sqrt{2}+1}{4}$$

Combine -2\sqrt{2}x and 2\sqrt{2}x to get 0.

$$\frac{4x^{2}+2x-\left(\sqrt{2}\right)^{2}+\sqrt{2}+2x-\sqrt{2}+1}{4}$$

The square of \sqrt{2} is 2.

$$\frac{4x^{2}+2x-2+\sqrt{2}+2x-\sqrt{2}+1}{4}$$

Combine 2x and 2x to get 4x.

$$\frac{4x^{2}+4x-2+\sqrt{2}-\sqrt{2}+1}{4}$$

Combine \sqrt{2} and -\sqrt{2} to get 0.

$$\frac{4x^{2}+4x-2+1}{4}$$

Add -2 and 1 to get -1.

$$\frac{4x^{2}+4x-1}{4}$$

Divide each term of 4x^{2}+4x-1 by 4 to get -\frac{1}{4}+x+x^{2}.

$$-\frac{1}{4}+x+x^{2}$$

Factor

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