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.

This article provides a comprehensive overview of all algebra formulas taught from Class 9 through Class 12.

Here are some of the most common algebra formulas:

Algebraic Identities Formula

Some important algebraic identities are:

Algebraic Properties

Various algebraic properties are mentioned below:

Commutative Property

Commutative Property is added in the table below:

Associative Property

Associative Property is added in the table below:

Distributive Property

Distributive Property is added in the table below:

Identity Element

Identity Element property is added in the table below:

Inverse Element

Inverse Element property is added in the table below:

Formulas in Algebra - PDF Download

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 8

Algebra formulas for class 8 are discussed below in this article. For three variables a, b, and c, the various algebraic formulas are:

• (a + b)² = a² + 2ab + b²
• (a - b)² = a² - 2ab + b²
• (a + b)(a - b) = a² - b²
• (a + b)³ = a³ + 3a²b + 3ab² + b³
• (a - b)³ = a³ - 3a²b + 3ab² - b³
• a³ + b³ = (a + b)(a² - ab + b²)
• a³ - b³ = (a - b)(a² + ab + b²)
• (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

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. These formulas are discussed below,

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

Exponents are ways to represent higher powers. 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

Articles Related to Algebraic Formulas:

• Algebra in Maths
• Basic Math Formulas
• What is Arithmetic?
• Algebraic Identities of Polynomials

Solved Question on Algebra Formulas

Question 1: Find out the value of the term, (2x + 3)² using algebraic formulae.

Using algebraic formula,

(a + b)² = a² + b² + 2ab

(2x + 3)² = (2x)² + 3² + 2 × 2x × 3

(2x + 3)² = 4x² + 9 + 12x

Question 2: Find out the value of the term, (5x - 3y)² using algebraic formulae.

Using algebraic formula,

(a - b)² = a² + b² - 2ab

(5x - 3y)² = (5x)² + (3y)² - 2 × 5x × 3y

(5 - 3)² = 25x² + 9y² - 30xy

Question 3: Find out the value of, 105 × 95 using algebraic formulae.

Using algebraic formula,

(a + b)(a - b) = a² - b²

105 × 95 = (100 + 5)(100 - 5)

= 100² - 5²

= 10000 - 25

= 9975

Question 4: Find the roots of the quadratic equation x² + 6x + 8=0 using algebra formulas for quadratic equations.

Given quadratic equation is x² + 6x + 8 = 0

Comparing above equation with ax²+bx+c=0, a=1, b=6, c=8 

Substituting the values in the quadratic formula we get,

x = [−b ± √(b² − 4ac)] / 2a

= [−6 ± √(6² − 4(1)(8))] / 2(1)

= [−6 ± √(36 − 4(1)(8))] / 2

= [−6 ± √(36 − 32)] / 2

= [−6 ± √4] / 2

= (-6 + 2) / 2 and (-6 - 2) / 2

= -4/2 and -8/2 = -2 and -4

Thus, the values of x are -2 and -4

Practice Problems - Algebra Formulas

Question 1. Find the value of x if (2x + 3)² = 49.

Question 2. Solve for y if (4y − 5)² = 121

Question 3. Determine the value of a in the equation (3a+2b)² = 144

Question 4. Solve for c if (c+3)³ = 512

Question 5. Fnd n if log⁡₇(a³) = 6 and log⁡₇(a) = 2.

Question 6. Simplify the expression using algebraic identities:(x+3)^2-(x-2)^2.

Question 7. Factorize the expression: x^3-8.

Question 8. Expand and simplify: (2a+3b)^3.

Question 9. Simplify the expression using logarithmic properties: \log_2{(16)}+\log_2{(8)}.

Question 10. Determine the 5th term of the geometric sequence where the first term a=3 and the common ratio r=2. the

Conclusion

Algebraic formulas are extremely helpful in simplifying and solving a variety of mathematical problems. They help us manipulate algebraic expressions in an organized way when handling polynomials, solving equations involving quadratics or logs, or dealing with sequences. To advance in mathematics, it is essential to know and be able to use these formulas because they serve as the basis for advanced topics as well as applications across disciplines. By frequently practicing with and applying them, one can enhance their problem-solving skills while developing a better understanding of algebraic concepts.

Summarizing Algebra Formulas

All the important algebra formulas are

• (a + b)² = a² + 2ab + b²
• (a - b)² = a² - 2ab + b²
• (a + b)(a - b) = a² - b²
• (a + b)³ = a³ + 3a²b + 3ab² + b³
• (a - b)³ = a³ - 3a²b + 3ab² - b³
• a³ + b³ = (a + b)(a² - ab + b²)
• a³ - b³ = (a - b)(a² + ab + b²)
• (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
• 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
• 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
• nth term of a AP, a_n = a + (n - 1) d
• Sum of the first n terms, S_n = n/2 [2a + (n - 1) d]
• nth term of a GP, 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)