Algebra Formulas - List of all Algebra Formulas

Algebra formulas are mathematical expressions that help solve problems involving variables and constants. They often represent relationships between quantities and can be used to simplify calculations.

Algebraic Identities Formula

Some important algebraic identities are:

Basic Formulas in Algebra

Basic algebra formulas help us to solve the fundamental and complicated mathematical problems. Various basic formulas are:

• a² – b² = (a – b)(a + b)
• (a + b)² = a² + 2ab + b²
• a² + b² = (a + b)² – 2ab
• (a – b)² = a² – 2ab + b²
• (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 ; (a + b)³ = 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 + b⁴
• a⁴ – b⁴ = (a – b)(a + b)(a² + b²)
• a⁵ – b⁵ = (a – b)(a⁴ + a³b + a²b² + ab³ + b⁴)
• aⁿ – bⁿ = (a – b)(a^n-1 + a^n-2b+…+ b^n-2a + b^n-1)
• (a^m)(aⁿ) = a^m+n ; (ab)^m = a^mb^m ; (a^m)ⁿ = a^mn

Algebra Formulas for Class 9

For class 9 logarithm formulas are very useful. They are helpful for the computation of highly complex problems of multiplication and division. The exponential form of 3² = 9 can easily be transformed into logarithmic form as log₃ 9 = 2.

Also, complex multiplication and division can easily be converted to addition and subtraction by following the logarithmic formulas. 

Important log algebraic formulas that are most commonly used are discussed below:

• log_a (xy) = log_a x + log_a y
• log_a (x/y) = log_a x - log_a y
• log_a x^m = m log_a x
• log_a a = 1
• log_a 1 = 0

Algebra Formulas for Class 10

"Quadratic Formula” is an important algebra equation and formulas that is introduced to students in class 10. It is used for solving general quadratic equations. The general form of any quadratic equation is ax² + bx + c = 0, where x is the variable a, b are coefficients and c is constant. There are two ways of solving this quadratic equation. 

• Solution using Algebraic Method
• Using Quadratic Formula

Other important formulas used in class 10 are

Formulas for Arithmetic Sequence

For any given arithmetic sequence {a, a + d, a + 2d, ...}

• nth term, a_n = a + (n - 1) d
• Sum of the first n terms, S_n = n/2 [2a + (n - 1) d]

Formulas for Geometric Sequences

For any given geometric sequence {a, ar, ar², ...}

• nth term, a_n = a rⁿ - 1
• Sum of the first n terms, S_n = a (1 - rⁿ) / (1 - r)
• Sum of infinite terms when r<1, S = a / (1 - r)

Algebra Formulas for Class 11

Algebra Formulas for Class 11, which are mostly used are formulas of permutations and combinations. If different arrangements of r things from the n available things are required then permutation formulas are used, whereas combinations formulas are used for finding the different groups of r things from n available things. 

The important permutation and combination formulas are,

Difference of powers

• If n is even, then: aⁿ – bⁿ = (a – b)(a^n-1 + a^n-2b +a^n-3b² + …+ ab^n-2 + b^n-1)
• If n is odd, then: aⁿ + bⁿ = (a + b)(a^n-1 – a^n-2b  +a^n-3b² – …- ab^n-2 + b^n-1)

Exponent and Power

• (a^m )(aⁿ ) = a^m+n 
• (ab)^m = a^mb^m
• (a^m)ⁿ = a^mn

Factorial Formula: n! = n × (n - 1) × (n - 2) × ... × 3 × 2 × 1

Permutation Formulas: ⁿP_r = n! / (n-r)!

Combination Formula: ⁿC_r = n!/[r!(n−r)!]

Note: Binomial Theorem is another formula that is of the utmost importance for students in class 11.

Algebra Formulas for Class 12

The important formulas for students in class 12 include vector algebra formulas:

Take any three vectors, a, b, and c then,

• For vector a = x i+y j+z k, then magnitude of |a| =√(x²+y²+z²).
• Unit vector along a is a / |a|
• Dot product of two vectors a and b is defined as a ⋅ b = |a| |b| cos θ,is
  where θ is the angle between the vectors a and b.
• Important unit vectors are \hat{i} = [1, 0, 0], \quad \hat{j} = [0, 1, 0], \quad \hat{k} = [0, 0, 1]
• Directional angles l = cos 𝞪, m = cos 𝞫, n = cos 𝛄, are called directional angles of the vector \cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1
• In Vector Addition\vec{a} + \vec{b} = \vec{b} + \vec{a}, \vec{a} + (\vec{b} + \vec{c}) = (\vec{a} + \vec{b}) + \vec{c}, k(\vec{a} + \vec{b}) = k\vec{a} + k\vec{b}
• Cross product of vectors a and b is defined as a × b = |a| |b| sin θ
  where θ is the angle between the vectors a and b.
• Scalar Triple Product of three vectors a, b, and c is given by [a b c ] = a ⋅ (b × c) = (a × b) ⋅ c.

Law of Exponents

Law of Exponents are used to solve problems with the higher power. Some of the common laws of exponents with the same bases having different powers, and different bases having the same power, are useful to solve complex exponential terms.

The higher exponential values can be easily solved without any expansion of the exponential terms. These exponential laws are further useful to derive some of the logarithmic laws.

• a^m× aⁿ = a^{m + n}
• a^m/aⁿ = a^{m - n}
• (a^m)ⁿ = a^mn
• (ab)^m = a^m× b^m
• a⁰ = 1
• a^-m = 1/a^m

Practice Questions on Algebra