Basic Algebra/Solving Equations/Formulas

Basic Formulas

 * 1) $$(a+b)^2=a^2+2ab+b^2$$
 * 2) $$(a-b)^2=a^2-2ab+b^2$$
 * 3) $$a^2-b^2=(a+b)(a-b)$$
 * 4) $$(a+b)^3=a^3+3a^2b+3ab^2+b^3$$
 * 5) $$(a-b)^3=a^3-3a^2b+3ab^2-b^3$$
 * 6) $$a^3+b^3=(a+b)(a^2-ab+b^2)$$
 * 7) $$a^3-b^3=(a-b)(a^2+ab+b^2)$$
 * 8) $$(a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca$$
 * 9) $$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$$

Derivation (1)
$$(a+b)^2$$

$$=(a+b)(a+b)$$

$$=a(a+b)+b(a+b)$$

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

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

Derivation (2)
$$(a-b)^2$$

$$=(a-b)(a-b)$$

$$=a(a-b)-b(a-b)$$

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

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

Derivation (3)
$$a^2-b^2$$

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

$$=a(a+b)-b(a+b)$$

$$=(a+b)(a-b)$$

Derivation (4)
$$(a+b)^3$$

$$=(a+b)\times (a+b)^2$$

$$=(a+b)\left (a^2+2ab+b^2\right )$$

$$=a\left (a^2+2ab+b^2\right )+b\left (a^2+2ab+b^2\right )$$ $$=a^3+2a^2b+ab^2+a^2b+2ab^2+b^3$$

$$=a^3+3a^2b+3ab^2+b^3$$

Derivation (5)
$$(a^3-b^3)$$

$$=(a-b)\times (a-b)^2$$

$$=(a-b)\left (a^2-2ab+b^2\right )$$

$$=a\left (a^2-2ab+b^2\right )-b\left (a^2-2ab+b^2\right )$$

$$=a^3-2a^2b+ab^2-a^2b+2ab^2-b^3$$

$$=a^3-3a^2b+3ab^2-b^3$$