Solve 7(x-2^{-1})(x+2^{-1})-x^2]^-2=(x-sqrt{2})^2+(x+sqrt{2})^2+2x

$$7 ( x - 2 ^ { - 1 } ) ( x + 2 ^ { - 1 } ) - x ^ { 2 } ] ^ { - 2 } = ( x - \sqrt { 2 } ) ^ { 2 } + ( x + \sqrt { 2 } ) ^ { 2 } + 2 x$$

Evaluate

$$6x^{2}-\frac{7}{4}$$

Solution Steps

Calculate 2 to the power of -1 and get \frac{1}{2}.

$$7\left(x-\frac{1}{2}\right)\left(x+2^{-1}\right)-x^{2}$$

Calculate 2 to the power of -1 and get \frac{1}{2}.

$$7\left(x-\frac{1}{2}\right)\left(x+\frac{1}{2}\right)-x^{2}$$

Use the distributive property to multiply 7 by x-\frac{1}{2}.

$$\left(7x-\frac{7}{2}\right)\left(x+\frac{1}{2}\right)-x^{2}$$

Use the distributive property to multiply 7x-\frac{7}{2} by x+\frac{1}{2} and combine like terms.

$$7x^{2}-\frac{7}{4}-x^{2}$$

Combine 7x^{2} and -x^{2} to get 6x^{2}.

$$6x^{2}-\frac{7}{4}$$

Expand

$$6x^{2}-\frac{7}{4}$$

Solution Steps

Calculate 2 to the power of -1 and get \frac{1}{2}.

$$7\left(x-\frac{1}{2}\right)\left(x+2^{-1}\right)-x^{2}$$

Calculate 2 to the power of -1 and get \frac{1}{2}.

$$7\left(x-\frac{1}{2}\right)\left(x+\frac{1}{2}\right)-x^{2}$$

Use the distributive property to multiply 7 by x-\frac{1}{2}.

$$\left(7x-\frac{7}{2}\right)\left(x+\frac{1}{2}\right)-x^{2}$$

Use the distributive property to multiply 7x-\frac{7}{2} by x+\frac{1}{2} and combine like terms.

$$7x^{2}-\frac{7}{4}-x^{2}$$

Combine 7x^{2} and -x^{2} to get 6x^{2}.

$$6x^{2}-\frac{7}{4}$$