Solve x^2-3=(x-sqrt{3})(x+sqrt{3})

$$x ^ { 2 } - 3 = ( x - \sqrt { 3 } ) ( x + \sqrt { 3 } )$$

Solve for x (complex solution)

$$x\in \mathrm{C}$$

Solution Steps

Consider \left(x-\sqrt{3}\right)\left(x+\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

The square of \sqrt{3} is 3.

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

Subtract x^{2} from both sides.

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

Combine x^{2} and -x^{2} to get 0.

$$-3=-3$$

Compare -3 and -3.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{C}$$

Solve for x

$$x\in \mathrm{R}$$

Solution Steps

Consider \left(x-\sqrt{3}\right)\left(x+\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

The square of \sqrt{3} is 3.

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

Subtract x^{2} from both sides.

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

Combine x^{2} and -x^{2} to get 0.

$$-3=-3$$

Compare -3 and -3.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{R}$$