Solve left(x-1+sqrt{3}right)left(x-1-sqrt{3}right)

$$\left( x-1+ \sqrt{ 3 } \right) \left( x-1- \sqrt{ 3 } \right)$$

Evaluate

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

Solution Steps

Apply the distributive property by multiplying each term of x-1+\sqrt{3} by each term of x-1-\sqrt{3}.

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

Combine -x and -x to get -2x.

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

Combine -x\sqrt{3} and \sqrt{3}x to get 0.

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

Combine \sqrt{3} and -\sqrt{3} to get 0.

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

The square of \sqrt{3} is 3.

$$x^{2}-2x+1-3$$

Subtract 3 from 1 to get -2.

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

Differentiate w.r.t. x

$$2\left(x-1\right)$$

Steps Using Derivative Rule for Sum

Apply the distributive property by multiplying each term of x-1+\sqrt{3} by each term of x-1-\sqrt{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-x-x\sqrt{3}-x+1+\sqrt{3}+\sqrt{3}x-\sqrt{3}-\left(\sqrt{3}\right)^{2})$$

Combine -x and -x to get -2x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-2x-x\sqrt{3}+1+\sqrt{3}+\sqrt{3}x-\sqrt{3}-\left(\sqrt{3}\right)^{2})$$

Combine -x\sqrt{3} and \sqrt{3}x to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-2x+1+\sqrt{3}-\sqrt{3}-\left(\sqrt{3}\right)^{2})$$

Combine \sqrt{3} and -\sqrt{3} to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-2x+1-\left(\sqrt{3}\right)^{2})$$

The square of \sqrt{3} is 3.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-2x+1-3)$$

Subtract 3 from 1 to get -2.

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

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}.

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

Subtract 1 from 2.

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

Subtract 1 from 1.

$$2x^{1}-2x^{0}$$

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

$$2x-2x^{0}$$

For any term t except 0, t^{0}=1.

$$2x-2$$