Algebra Formulas

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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

ALGEBRA FORMULAS

Arithmetic Properties

( 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 = a 3 + b 3 + 3 a b ( a + b ) ( a − b ) 3 = a 3 − 3 a 2 b + 3 a b 2 − b 3 = a 3 − b 3 − 3 a b ( a − b ) a 3 − b 3 = ( a − b ) ( a 2 + a b + b 2 ) a 3 + b 3 = ( a + b ) ( a 2 − a b + b 2 ) ( 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 ) ( a 2 + b 2 ) a 5 − b 5 = ( a − b ) ( a 4 + a 3 b + a 2 b 2 + a b 3 + b 4 ) ( 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 = a 3 + b 3 + 3 a b ( a + b ) ( a − b ) 3 = a 3 − 3 a 2 b + 3 a b 2 − b 3 = a 3 − b 3 − 3 a b ( a − b ) a 3 − b 3 = ( a − b ) a 2 + a b + b 2 a 3 + b 3 = ( a + b ) a 2 − a b + b 2 ( 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 ) a 2 + b 2 a 5 − b 5 = ( a − b ) a 4 + a 3 b + a 2 b 2 + a b 3 + b 4 {:[(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}(a+b)2=a2+2ab+b2(a−b)2=a2−2ab+b2a2+b2=(a+b)2−2aba2−b2=(a+b)(a−b)(a+b+c)2=a2+b2+c2+2ab+2bc+2ca(a+b−c)2=a2+b2+c2+2ab−2bc−2ca(a−b−c)2=a2+b2+c2−2ab+2bc−2ca(a+b)3=a3+3a2b+3ab2+b3=a3+b3+3ab(a+b)(a−b)3=a3−3a2b+3ab2−b3=a3−b3−3ab(a−b)a3−b3=(a−b)(a2+ab+b2)a3+b3=(a+b)(a2−ab+b2)(a+b)4=a4+4a3b+6a2b2+4ab3+b4(a−b)4=a4−4a3b+6a2b2−4ab3+b4a4−b4=(a+b)(a−b)(a2+b2)a5−b5=(a−b)(a4+a3b+a2b2+ab3+b4)

If n n n\mathbf{n}n is a Natural Number:

a n − b n = ( a − b ) ( a n − 1 + a n − 2 b + ⋯ + a b n − 2 + b n − 1 ) a n − b n = ( a − b ) a n − 1 + a n − 2 b + ⋯ + a b n − 2 + b n − 1 a^(n)-b^(n)=(a-b)(a^(n-1)+a^(n-2)b+cdots+ab^(n-2)+b^(n-1))a^{n}-b^{n}=(a-b)\left(a^{n-1}+a^{n-2} b+\cdots+a b^{n-2}+b^{n-1}\right)an−bn=(a−b)(an−1+an−2b+⋯+abn−2+bn−1)
○ If n n n\mathbf{n}n is Even ( n = 2 k k n = 2 k k n=2kk\mathbf{n}=\mathbf{2 k} \mathbf{k}n=2kk :
a n + b n = ( a + b ) ( a n − 1 − a n − 2 b + ⋯ + a b n − 2 − b n − 1 ) a n + b n = ( a + b ) a n − 1 − a n − 2 b + ⋯ + a b n − 2 − b n − 1 a^(n)+b^(n)=(a+b)(a^(n-1)-a^(n-2)b+cdots+ab^(n-2)-b^(n-1))a^{n}+b^{n}=(a+b)\left(a^{n-1}-a^{n-2} b+\cdots+a b^{n-2}-b^{n-1}\right)an+bn=(a+b)(an−1−an−2b+⋯+abn−2−bn−1)

If n n n\mathbf{n}n is Odd ( n = 2 k + 1 ) ( n = 2 k + 1 ) (n=2k+1)(\mathbf{n}=\mathbf{2} \mathbf{k}+\mathbf{1})(n=2k+1) :

a n + b n = ( a + b ) ( a n − 1 − a n − 2 b + a n − 3 b 2 + ⋯ − a b n − 2 + b n − 1 ) a n + b n = ( a + b ) a n − 1 − a n − 2 b + a n − 3 b 2 + ⋯ − a b n − 2 + b n − 1 a^(n)+b^(n)=(a+b)(a^(n-1)-a^(n-2)b+a^(n-3)b^(2)+cdots-ab^(n-2)+b^(n-1))a^{n}+b^{n}=(a+b)\left(a^{n-1}-a^{n-2} b+a^{n-3} b^{2}+\cdots-a b^{n-2}+b^{n-1}\right)an+bn=(a+b)(an−1−an−2b+an−3b2+⋯−abn−2+bn−1)
a n m = a m n a n m = a m n root(m)(root(n)(a))=root(mn)(a)\sqrt[m]{\sqrt[n]{a}}=\sqrt[m n]{a}anm=amn ∘ ∘ @\circ∘ a m n = a m n a m n = a m n root(n)(a^(m))=a^((m)/(n))\sqrt[n]{a^{m}}=a^{\frac{m}{n}}amn=amn
a b n = a n b n a b n = a n b n root(n)(ab)=root(n)(a)root(n)(b)\sqrt[n]{a b}=\sqrt[n]{a} \sqrt[n]{b}abn=anbn ∘ ∘ @\circ∘ a n n = a a n n = a root(n)(a^(n))=a\sqrt[n]{a^{n}}=aann=a, if n n nnn is odd
a b n = a n b n a b n = a n b n root(n)((a)/(b))=(root(n)(a))/(root(n)(b))\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}abn=anbn ∘ ∘ @\circ∘ a n n = | a | a n n = | a | root(n)(a^(n))=|a|\sqrt[n]{a^{n}}=|a|ann=|a|, if n n nnn is even
root(m)(root(n)(a))=root(mn)(a) @ root(n)(a^(m))=a^((m)/(n)) root(n)(ab)=root(n)(a)root(n)(b) @ root(n)(a^(n))=a, if n is odd root(n)((a)/(b))=(root(n)(a))/(root(n)(b)) @ root(n)(a^(n))=|a|, if n is even| $\sqrt[m]{\sqrt[n]{a}}=\sqrt[m n]{a}$ | $\circ$ | $\sqrt[n]{a^{m}}=a^{\frac{m}{n}}$ | | :---: | :---: | :---: | | $\sqrt[n]{a b}=\sqrt[n]{a} \sqrt[n]{b}$ | $\circ$ | $\sqrt[n]{a^{n}}=a$, if $n$ is odd | | $\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}$ | $\circ$ | $\sqrt[n]{a^{n}}=\|a\|$, if $n$ is even |
a 2 n − b 2 n = ( a n + b n ) ( a n − b n ) a 2 n − b 2 n = a n + b n a n − b n a^(2n)-b^(2n)=(a^(n)+b^(n))(a^(n)-b^(n))a^{2 n}-b^{2 n}=\left(a^{n}+b^{n}\right)\left(a^{n}-b^{n}\right)a2n−b2n=(an+bn)(an−bn)
( 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+cdots)^(2)=a^(2)+b^(2)+c^(2)+cdots+2ab+2bc+2ca+cdots(a+b+c+\cdots)^{2}=a^{2}+b^{2}+c^{2}+\cdots+2 a b+2 b c+2 c a+\cdots(a+b+c+⋯)2=a2+b2+c2+⋯+2ab+2bc+2ca+⋯
Associative: Commutative: Distributive: a + b = b + a and a b = b a a ( b + c ) = a b + a c a b + a c a = b + c , a ≠ 0 a b + c d = a d + b c b d a b − c d = a d − b c b d a b c = a c b a + b c = a c + b c a b c = a b c Associative: Commutative: Distributive: a + b = b + a and a b = b a a ( b + c ) = a b + a c a b + a c a = b + c , a ≠ 0 a b + c d = a d + b c b d a b − c d = a d − b c b d a b c = a c b a + b c = a c + b c a b c = a b c {:[" Associative: "],[" Commutative: "],[" Distributive: "],[a+b=b+a" and "ab=ba],[a(b+c)=ab+ac],[(ab+ac)/(a)=b+c","a!=0],[(a)/(b)+(c)/(d)=(ad+bc)/(bd)],[{:[(a)/(b)-(c)/(d)=(ad-bc)/(bd)],[(a)/((b)/(c))=(ac)/(b)],[(a+b)/(c)=(a)/(c)+(b)/(c)],[((a)/(b))/(c)=(a)/(bc)]:}]:}\begin{aligned} & \text { Associative: } \\ & \text { Commutative: } \\ & \text { Distributive: } \\ & a+b=b+a \text { and } a b=b a \\ & a(b+c)=a b+a c \\ & \frac{a b+a c}{a}=b+c, a \neq 0 \\ & \frac{a}{b}+\frac{c}{d}=\frac{a d+b c}{b d} \\ & \begin{array}{l} \frac{a}{b}-\frac{c}{d}=\frac{a d-b c}{b d} \\ \frac{a}{\frac{b}{c}}=\frac{a c}{b} \\ \frac{\mathrm{a}+\mathrm{b}}{\mathrm{c}}=\frac{a}{c}+\frac{b}{c} \\ \frac{\frac{a}{b}}{c}=\frac{a}{b c} \end{array} \end{aligned} Associative: Commutative: Distributive: a+b=b+a and ab=baa(b+c)=ab+acab+aca=b+c,a≠0ab+cd=ad+bcbdab−cd=ad−bcbdabc=acba+bc=ac+bcabc=abc

Quadratic Equation

For the Equation: a x 2 + b x + c = 0 a x 2 + b x + c = 0 ax^(2)+bx+c=0a x^{2}+b x+c=0ax2+bx+c=0
Solution: x = − b ± b 2 − 4 a c 2 a x = − b ± b 2 − 4 a c 2 a x=(-b+-sqrt(b^(2)-4ac))/(2a)x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}x=−b±b2−4ac2a

Properties of Inequalities

If a < b a < b a < ba<ba<b then a + c < b + c a + c < b + c a+c < b+ca+c<b+ca+c<b+c and a − c < b − c a − c < b − c a-c < b-ca-c<b-ca−c<b−c
If a < b a < b a < ba<ba<b and c > 0 c > 0 c > 0c>0c>0 then a c < b c a c < b c ac < bca c<b cac<bc and a / c = b / c a / c = b / c a//c=b//ca / c=b / ca/c=b/c
If a < b a < b a < ba<ba<b and c < 0 c < 0 c < 0c<0c<0 then a c > b c a c > b c ac > bca c>b cac>bc and a / c > b / c a / c > b / c a//c > b//ca / c>b / ca/c>b/c

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