Solve (x-sqrt{2})(x+sqrt{2})-4(x-1)(x+1)

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

Evaluate

$$2-3x^{2}$$

Solution Steps

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}.

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

The square of \sqrt{2} is 2.

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

Use the distributive property to multiply -4 by x-1.

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

Apply the distributive property by multiplying each term of -4x+4 by each term of x+1.

$$x^{2}-2-4x^{2}-4x+4x+4$$

Combine -4x and 4x to get 0.

$$x^{2}-2-4x^{2}+4$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$-3x^{2}-2+4$$

Add -2 and 4 to get 2.

$$-3x^{2}+2$$

Differentiate w.r.t. x

$$-6x$$

Steps Using Derivative Rule for Sum

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}.

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

The square of \sqrt{2} is 2.

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

Use the distributive property to multiply -4 by x-1.

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

Apply the distributive property by multiplying each term of -4x+4 by each term of x+1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-2-4x^{2}-4x+4x+4)$$

Combine -4x and 4x to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-2-4x^{2}+4)$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-3x^{2}-2+4)$$

Add -2 and 4 to get 2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-3x^{2}+2)$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$2\left(-3\right)x^{2-1}$$

Multiply 2 times -3.

$$-6x^{2-1}$$

Subtract 1 from 2.

$$-6x^{1}$$

For any term t, t^{1}=t.

$$-6x$$