Basic Algebra/Solving Equations/Formulas

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Basic Formulas

$(a+b)^{2}=a^{2}+2ab+b^{2}$
$(a-b)^{2}=a^{2}-2ab+b^{2}$
$a^{2}-b^{2}=(a+b)(a-b)$
$(a+b)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3}$
$(a-b)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3}$
$a^{3}+b^{3}=(a+b)(a^{2}-ab+b^{2})$
$a^{3}-b^{3}=(a-b)(a^{2}+ab+b^{2})$
$(a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2ab+2bc+2ca$
$a^{3}+b^{3}+c^{3}-3abc=(a+b+c)(a^{2}+b^{2}+c^{2}-ab-bc-ca)$

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)

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

Derivation (5)

$(a^{3}-b^{3})$
$=(a-b)\times (a-b)^{2}$
$=(a-b)\left(a^{2}-2ab+b^{2}\right)$
$=a\left(a^{2}-2ab+b^{2}\right)-b\left(a^{2}-2ab+b^{2}\right)$
$=a^{3}-2a^{2}b+ab^{2}-a^{2}b+2ab^{2}-b^{3}$
$=a^{3}-3a^{2}b+3ab^{2}-b^{3}$

Derivation (6)

Derivation (7)

Derivation (8)

Derivation (9)

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