Algebraic Identities (radical, rational, conditional)

$$\sqrt{x \pm \sqrt{y}} =$$

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

$$\sqrt{a + b + 2\sqrt{ab}} =$$

$$\sqrt{a} + \sqrt{b}$$

$$\sqrt{a + b - 2\sqrt{ab}} =$$

$$\sqrt{a} - \sqrt{b}$$

$$\frac{a - b}{\sqrt[n]{a} - \sqrt[n]{b}} =$$

$$\underset{k=0}{\overset{n-1}{\sum}} \sqrt[n]{a^{(n-k)-1} b^{k}}$$

$$\frac{1}{(x-a)(x-b)} =$$

$$\frac{1}{a-b}\left( \frac{1}{x-a} - \frac{1}{x-b} \right)$$

$$\frac{1}{(a-b)(a-c)} + \frac{1}{(b-c)(b-a)}+ \frac{1}{(c-a)(c-b)} =$$

$$0$$

$$\frac{1}{(a-b)^2} + \frac{1}{(b-c)^2} + \frac{1}{(c-a)^2} =$$

$$\left( \frac{1}{a-b} + \frac{1}{b-c} + \frac{1}{c-a} \right)^2$$

$$\frac{a^3 + b^3}{a + b} =$$

$$a^2 - ab + b^2$$

$$\frac{a^3 - b^3}{a - b} =$$

$$a^2 + ab + b^2$$

$$\forall k \in \mathcal{I}_n \space \left( A_k = \underset{j \in \mathcal{I}_n \setminus \{k\}}{\prod} \frac{1}{a_k - a_j} \right) \implies \frac{1}{\underset{k=1}{\overset{n}{\prod}} (x - a_k)} =$$

$$\underset{k=1}{\overset{n}{\sum}} \frac{A_k}{x - a_k}$$

$$a + b + c = 0 \implies a^2 + b^2 + c^2 =$$

$$-2(ab + bc + ca)$$

$$a + b + c = 0 \implies a^4 + b^4 + c^4 =$$

$$2(a^2 b^2 + b^2 c^2 + c^2 a^2)$$

$$a + b + c = 0 \implies a^4 + b^4 + c^4 =$$

$$\frac{1}{2}(a^2 + b^2 + c^2)^2$$

$$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{a + b + c} \implies (a + b)(b + c)(c + a) =$$

$$0$$

$$ab + bc + ca = abc \implies \frac{a + b}{ab} + \frac{b + c}{bc} + \frac{c + a}{ca} =$$

$$2$$