Solve 2(x-frac{2+sqrt{6}}{2}y)(x-frac{2-sqrt{6}}{2}y)

$$2 ( x - \frac { 2 + \sqrt { 6 } } { 2 } y ) ( x - \frac { 2 - \sqrt { 6 } } { 2 } y )$$

Evaluate

$$2x^{2}-4xy-y^{2}$$

Short Solution Steps

Express \frac{2+\sqrt{6}}{2}y as a single fraction.

$$2\left(x-\frac{\left(2+\sqrt{6}\right)y}{2}\right)\left(x-\frac{2-\sqrt{6}}{2}y\right)$$

Use the distributive property to multiply 2+\sqrt{6} by y.

$$2\left(x-\frac{2y+\sqrt{6}y}{2}\right)\left(x-\frac{2-\sqrt{6}}{2}y\right)$$

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

$$2\left(\frac{2x}{2}-\frac{2y+\sqrt{6}y}{2}\right)\left(x-\frac{2-\sqrt{6}}{2}y\right)$$

Since \frac{2x}{2} and \frac{2y+\sqrt{6}y}{2} have the same denominator, subtract them by subtracting their numerators.

$$2\times \frac{2x-\left(2y+\sqrt{6}y\right)}{2}\left(x-\frac{2-\sqrt{6}}{2}y\right)$$

Do the multiplications in 2x-\left(2y+\sqrt{6}y\right).

$$2\times \frac{2x-2y-\sqrt{6}y}{2}\left(x-\frac{2-\sqrt{6}}{2}y\right)$$

Express \frac{2-\sqrt{6}}{2}y as a single fraction.

$$2\times \frac{2x-2y-\sqrt{6}y}{2}\left(x-\frac{\left(2-\sqrt{6}\right)y}{2}\right)$$

Use the distributive property to multiply 2-\sqrt{6} by y.

$$2\times \frac{2x-2y-\sqrt{6}y}{2}\left(x-\frac{2y-\sqrt{6}y}{2}\right)$$

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

$$2\times \frac{2x-2y-\sqrt{6}y}{2}\left(\frac{2x}{2}-\frac{2y-\sqrt{6}y}{2}\right)$$

Since \frac{2x}{2} and \frac{2y-\sqrt{6}y}{2} have the same denominator, subtract them by subtracting their numerators.

$$2\times \frac{2x-2y-\sqrt{6}y}{2}\times \frac{2x-\left(2y-\sqrt{6}y\right)}{2}$$

Do the multiplications in 2x-\left(2y-\sqrt{6}y\right).

$$2\times \frac{2x-2y-\sqrt{6}y}{2}\times \frac{2x-2y+\sqrt{6}y}{2}$$

Cancel out 2 and 2.

$$\left(2x-2y-\sqrt{6}y\right)\times \frac{2x-2y+\sqrt{6}y}{2}$$

Express \left(2x-2y-\sqrt{6}y\right)\times \frac{2x-2y+\sqrt{6}y}{2} as a single fraction.

$$\frac{\left(2x-2y-\sqrt{6}y\right)\left(2x-2y+\sqrt{6}y\right)}{2}$$

Apply the distributive property by multiplying each term of 2x-2y-\sqrt{6}y by each term of 2x-2y+\sqrt{6}y.

$$\frac{4x^{2}-4xy+2x\sqrt{6}y-4xy+4y^{2}-2\sqrt{6}y^{2}-2\sqrt{6}yx+2\sqrt{6}y^{2}-\left(\sqrt{6}\right)^{2}y^{2}}{2}$$

Combine -4xy and -4xy to get -8xy.

$$\frac{4x^{2}-8xy+2x\sqrt{6}y+4y^{2}-2\sqrt{6}y^{2}-2\sqrt{6}yx+2\sqrt{6}y^{2}-\left(\sqrt{6}\right)^{2}y^{2}}{2}$$

Combine 2x\sqrt{6}y and -2\sqrt{6}yx to get 0.

$$\frac{4x^{2}-8xy+4y^{2}-2\sqrt{6}y^{2}+2\sqrt{6}y^{2}-\left(\sqrt{6}\right)^{2}y^{2}}{2}$$

Combine -2\sqrt{6}y^{2} and 2\sqrt{6}y^{2} to get 0.

$$\frac{4x^{2}-8xy+4y^{2}-\left(\sqrt{6}\right)^{2}y^{2}}{2}$$

The square of \sqrt{6} is 6.

$$\frac{4x^{2}-8xy+4y^{2}-6y^{2}}{2}$$

Combine 4y^{2} and -6y^{2} to get -2y^{2}.

$$\frac{4x^{2}-8xy-2y^{2}}{2}$$

Divide each term of 4x^{2}-8xy-2y^{2} by 2 to get 2x^{2}-4xy-y^{2}.

$$2x^{2}-4xy-y^{2}$$

Factor

$$2x^{2}-4xy-y^{2}$$