# Algebra Formulas

Algebra is a fundamental branch of mathematics that deals with the manipulation of variables and equations. Mastering algebra formulas is crucial for solving a wide range of mathematical problems, from simple linear equations to complex systems of equations. Here are some of the most important algebra formulas you should know:

# ALGEBRA FORMULAS

## Arithmetic Properties

$\begin{array}{rl}& \left(a+b{\right)}^{2}={a}^{2}+2ab+{b}^{2}\\ & \left(a-b{\right)}^{2}={a}^{2}-2ab+{b}^{2}\\ & {a}^{2}+{b}^{2}=\left(a+b{\right)}^{2}-2ab\\ & {a}^{2}-{b}^{2}=\left(a+b\right)\left(a-b\right)\\ & \left(a+b+c{\right)}^{2}={a}^{2}+{b}^{2}+{c}^{2}+2ab+2bc+2ca\\ & \left(a+b-c{\right)}^{2}={a}^{2}+{b}^{2}+{c}^{2}+2ab-2bc-2ca\\ & \left(a-b-c{\right)}^{2}={a}^{2}+{b}^{2}+{c}^{2}-2ab+2bc-2ca\\ & \left(a+b{\right)}^{3}={a}^{3}+3{a}^{2}b+3a{b}^{2}+{b}^{3}\\ & \phantom{\rule{1em}{0ex}}={a}^{3}+{b}^{3}+3ab\left(a+b\right)\\ & \left(a-b{\right)}^{3}={a}^{3}-3{a}^{2}b+3a{b}^{2}-{b}^{3}\\ & \phantom{\rule{1em}{0ex}}={a}^{3}-{b}^{3}-3ab\left(a-b\right)\\ & {a}^{3}-{b}^{3}=\left(a-b\right)\left({a}^{2}+ab+{b}^{2}\right)\\ & {a}^{3}+{b}^{3}=\left(a+b\right)\left({a}^{2}-ab+{b}^{2}\right)\\ & \left(a+b{\right)}^{4}={a}^{4}+4{a}^{3}b+6{a}^{2}{b}^{2}+4a{b}^{3}+{b}^{4}\\ & \left(a-b{\right)}^{4}={a}^{4}-4{a}^{3}b+6{a}^{2}{b}^{2}-4a{b}^{3}+{b}^{4}\\ & {a}^{4}-{b}^{4}=\left(a+b\right)\left(a-b\right)\left({a}^{2}+{b}^{2}\right)\\ & {a}^{5}-{b}^{5}=\left(a-b\right)\left({a}^{4}+{a}^{3}b+{a}^{2}{b}^{2}+a{b}^{3}+{b}^{4}\right)\end{array}$$\begin{array}{r}\left(a+b{\right)}^{2}={a}^{2}+2ab+{b}^{2}\\ \left(aâˆ’b{\right)}^{2}={a}^{2}âˆ’2ab+{b}^{2}\\ {a}^{2}+{b}^{2}=\left(a+b{\right)}^{2}âˆ’2ab\\ {a}^{2}âˆ’{b}^{2}=\left(a+b\right)\left(aâˆ’b\right)\\ \left(a+b+c{\right)}^{2}={a}^{2}+{b}^{2}+{c}^{2}+2ab+2bc+2ca\\ \left(a+bâˆ’c{\right)}^{2}={a}^{2}+{b}^{2}+{c}^{2}+2abâˆ’2bcâˆ’2ca\\ \left(aâˆ’bâˆ’c{\right)}^{2}={a}^{2}+{b}^{2}+{c}^{2}âˆ’2ab+2bcâˆ’2ca\\ \left(a+b{\right)}^{3}={a}^{3}+3{a}^{2}b+3a{b}^{2}+{b}^{3}\\ \phantom{\rule{1em}{0ex}}={a}^{3}+{b}^{3}+3ab\left(a+b\right)\\ \left(aâˆ’b{\right)}^{3}={a}^{3}âˆ’3{a}^{2}b+3a{b}^{2}âˆ’{b}^{3}\\ \phantom{\rule{1em}{0ex}}={a}^{3}âˆ’{b}^{3}âˆ’3ab\left(aâˆ’b\right)\\ {a}^{3}âˆ’{b}^{3}=\left(aâˆ’b\right)\left({a}^{2}+ab+{b}^{2}\right)\\ {a}^{3}+{b}^{3}=\left(a+b\right)\left({a}^{2}âˆ’ab+{b}^{2}\right)\\ \left(a+b{\right)}^{4}={a}^{4}+4{a}^{3}b+6{a}^{2}{b}^{2}+4a{b}^{3}+{b}^{4}\\ \left(aâˆ’b{\right)}^{4}={a}^{4}âˆ’4{a}^{3}b+6{a}^{2}{b}^{2}âˆ’4a{b}^{3}+{b}^{4}\\ {a}^{4}âˆ’{b}^{4}=\left(a+b\right)\left(aâˆ’b\right)\left({a}^{2}+{b}^{2}\right)\\ {a}^{5}âˆ’{b}^{5}=\left(aâˆ’b\right)\left({a}^{4}+{a}^{3}b+{a}^{2}{b}^{2}+a{b}^{3}+{b}^{4}\right)\end{array}${:[(a+b)^(2)=a^(2)+2ab+b^(2)],[(a-b)^(2)=a^(2)-2ab+b^(2)],[a^(2)+b^(2)=(a+b)^(2)-2ab],[a^(2)-b^(2)=(a+b)(a-b)],[(a+b+c)^(2)=a^(2)+b^(2)+c^(2)+2ab+2bc+2ca],[(a+b-c)^(2)=a^(2)+b^(2)+c^(2)+2ab-2bc-2ca],[(a-b-c)^(2)=a^(2)+b^(2)+c^(2)-2ab+2bc-2ca],[(a+b)^(3)=a^(3)+3a^(2)b+3ab^(2)+b^(3)],[quad=a^(3)+b^(3)+3ab(a+b)],[(a-b)^(3)=a^(3)-3a^(2)b+3ab^(2)-b^(3)],[quad=a^(3)-b^(3)-3ab(a-b)],[a^(3)-b^(3)=(a-b)(a^(2)+ab+b^(2))],[a^(3)+b^(3)=(a+b)(a^(2)-ab+b^(2))],[(a+b)^(4)=a^(4)+4a^(3)b+6a^(2)b^(2)+4ab^(3)+b^(4)],[(a-b)^(4)=a^(4)-4a^(3)b+6a^(2)b^(2)-4ab^(3)+b^(4)],[a^(4)-b^(4)=(a+b)(a-b)(a^(2)+b^(2))],[a^(5)-b^(5)=(a-b)(a^(4)+a^(3)b+a^(2)b^(2)+ab^(3)+b^(4))]:}\begin{aligned} & (a+b)^{2}=a^{2}+2 a b+b^{2} \\ & (a-b)^{2}=a^{2}-2 a b+b^{2} \\ & a^{2}+b^{2}=(a+b)^{2}-2 a b \\ & a^{2}-b^{2}=(a+b)(a-b) \\ & (a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2 a b+2 b c+2 c a \\ & (a+b-c)^{2}=a^{2}+b^{2}+c^{2}+2 a b-2 b c-2 c a \\ & (a-b-c)^{2}=a^{2}+b^{2}+c^{2}-2 a b+2 b c-2 c a \\ & (a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3} \\ & \quad=a^{3}+b^{3}+3 a b(a+b) \\ & (a-b)^{3}=a^{3}-3 a^{2} b+3 a b^{2}-b^{3} \\ & \quad=a^{3}-b^{3}-3 a b(a-b) \\ & a^{3}-b^{3}=(a-b)\left(a^{2}+a b+b^{2}\right) \\ & a^{3}+b^{3}=(a+b)\left(a^{2}-a b+b^{2}\right) \\ & (a+b)^{4}=a^{4}+4 a^{3} b+6 a^{2} b^{2}+4 a b^{3}+b^{4} \\ & (a-b)^{4}=a^{4}-4 a^{3} b+6 a^{2} b^{2}-4 a b^{3}+b^{4} \\ & a^{4}-b^{4}=(a+b)(a-b)\left(a^{2}+b^{2}\right) \\ & a^{5}-b^{5}=(a-b)\left(a^{4}+a^{3} b+a^{2} b^{2}+a b^{3}+b^{4}\right) \end{aligned}