Algebra Formulas PDF: Complete Algebraic Identities & Formula Chart

Algebra is a fundamental branch of mathematics that uses variables to represent numbers. In school, algebraic formulas serve as the building blocks for advanced topics, including quadratic equations, polynomials, coordinate geometry, calculus, trigonometry, and probability. Understanding these identities is vital for solving complex problems efficiently. Download our comprehensive Algebra Formulas PDF and access a complete chart of algebraic identities all in one place.

Algebra Formula

Algebra is a cornerstone of mathematics, essential for solving problems across various disciplines like geometry, calculus, and statistics. By using variables such as X, Y, A, and B, algebraic formulas allow us to tackle intricate computations with speed and precision. This guide provides a collection of significant algebraic formulas and step-by-step solutions to help students consolidate their learning.

Algebra Formulas Example

Algebraic formulas are mathematical equations composed of variables, constants, and operators. These expressions utilize unknown variables like 'x' to represent values, which are solved through systematic simplification. These tools are indispensable for streamlining complex algebraic calculations.
For example,

(a+b)³ = a³ + 3a²b + 3ab² + b³

In the formula above, both sides represent the same algebraic value. Specifically, (a³ + 3a²b + 3ab² + b³) is the expanded form of the expression (a+b)³.

An algebraic identity is an equality that holds true for all possible values of the variables involved. It confirms that the left-hand side (LHS) is equivalent to the right-hand side (RHS). Mastery of these identities is key to solving for unknown variables. Below are some of the most commonly used algebraic identities.

Algebraic Identities Formula

• (a + b)² = a² + 2ab + b²
• (a – b)² = a² – 2ab + b²
• (a + b)(a – b) = a² – b²
• (x + a)(x + b) = x² + x(a + b) + ab

Algebra Formulas For Squares of Class 10

Here are the essential square-based formulas:
• 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

Algebra Formulas for Cube

Review the following algebraic formulas involving cubes:

• (a + b)³ = a³ + 3a²b + 3ab² + b³
• (a + b)³ = a³ + b³ + 3ab(a + b)
• (a – b)³ = a³ – 3a²b + 3ab² – b³
• (a – b)³ = a³ – b³ – 3ab(a – b)
• a³ – b³ = (a – b)(a² + ab + b²)
• a³ + b³ = (a + b)(a² – ab + b²)

Additional important algebra formulas include:
• (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)

Algebra Formula for Natural Numbers

Algebraic formulas for natural numbers are used to simplify operations involving counting numbers (starting from 1). Here, we denote 'n' as a natural number to apply these specific operational rules.

1. (aⁿ – bⁿ )= (a – b)(a^n-1 + a^n-2b+…+ b^n-2a + b^n-1)
2. ( aⁿ + bⁿ)= (a + b)(a^n-1 – a^n-2b +…+ b^n-2a – b^n-1) [ where n is even  , (n = k + 1)  ]
3. (aⁿ + bⁿ )= (a + b)(a^n-1 – a^n-2b +a^n-3b²…- b^n-2a + b^n-1) [ where n is odd , (n = 2k + 1)  ]

Laws of Exponents

In algebra, exponents represent repeated multiplication. For example, 3×3×3×3 is expressed as 3⁴, where 4 is the exponent of the base 3. These powers follow specific rules for addition, subtraction, and multiplication, which are effectively handled by standard algebraic formulas.

Algebra Formulas for Quadric equations

Quadratic equations are a primary topic in Class 9 and 10 mathematics. To determine the roots of these equations, we use specific foundational formulas.

For a quadratic equation in the form ax² + bx + c = 0, the solutions are calculated as follows:

From the standard quadratic formula, if the roots are α and β, then:
1. The equation is represented as (x − α)(x − β) = 0.
2. The sum of the roots (α + β) = -b/a, and the product (α × β) = c/a.

Algebra Formulas For Irrational Numbers

Specialized formulas are also required to solve equations involving irrational numbers.

1. √ab = √a √b
2. √a/b =√a / √b
3. ( √a +√b )  ( √a – √b ) = a-b
4. ( √a +√b )²= a + 2 √ab + b
5. ( a +√b )( a -√b )= a² – b

Algebra Formulas List/Chart

We have compiled a comprehensive list of all critical algebraic formulas. Students are encouraged to review these to solve complex algebraic equations with greater speed and accuracy.

	Important Formulas
1	a²– b² = (a – b)(a + b)
2	(a + b)²=  a²+ 2ab + b²
3	a²+ b²= (a + b)²– 2ab
4	(a – b)² = a²– 2ab+ b²
5	(a + b + c)² = a² + b² + c²+ 2ab + 2bc + 2ca
6	(a – b – c)² = a²+ b²+ c²– 2ab + 2bc – 2ca
7	(a + b)³ = a³+ 3a²b + 3ab²+ b³
8	(a + b)³ = a³ + b³ + 3ab(a + b)
9	(a – b)³= a³ – 3a²b + 3ab² – b³
10	(a – b)³= a³ – b³ – 3ab(a – b)
11	a³ – b³ = (a – b)(a²+ ab + b²)
12	a³ + b³ = (a + b)(a²– ab + b²)
13	(a + b)⁴= a⁴+ 4a³b + 6a²b² + 4ab³ + b²
14	(a – b)⁴= a4 – 4a³b + 6a²b² – 4ab³+ b⁴
15	a⁴ – b⁴= (a – b)(a + b)(a² + b²)
16	a⁵ – b⁵= (a – b)(a⁴ + a³b + a²b² + ab³+ b⁴)

Some commonly used Algebraic formulas in maths

Below, we provide the derivations for several frequently used algebra formulas to enhance your conceptual understanding.

a+b+c Whole Square Formula- (a+b+c)^2 Formula

The square of the sum of three terms can be expanded using the trinomial expansion formula:

(a+b+c)2=a2+b2+c2+2ab+2ac+2bc

Thus, (a+b+c)² is equal to the sum of the squares of each individual term and twice the product of the pairs of terms (2ab, 2ac, 2bc).

This expansion is valid for all values of a, b, and c.

a-b Whole Square

The square of the difference between two terms is expanded using the binomial square formula:

(a−b)2=a2−2ab+b2

Essentially, (a − b)² equals the square of the first term minus twice the product of the two terms, plus the square of the second term.

This rule holds true for any values of a and b.

a+b Whole Square

The square of the sum of two terms, (a + b), follows the classic binomial square expansion:

(a+b)2=a2+2ab+b2

Thus, (a + b)² equals the sum of the squares of the individual terms and twice their product.

This expansion is universally true for all values of a and b.

Implementation of Algebra All Formulas with Examples

Example 1: Find the value of 20² - 15²
Solution: We apply the formula a² - b² = (a+b)(a-b)
= (20+15)(20-15)
= 35 × 5
= 175 (Answer)

Example 2: If (x-y) = 2 and x² + y² = 20, find the values of x and y [where x, y > 0]
Solution: Since x² + y² = 20, we use (x-y)² + 2xy = 20
(2)² + 2xy = 20 => 2xy = 16 => xy = 8
Now, (x+y)² = (x-y)² + 4xy = 4 + 32 = 36
Thus, x+y = 6 (as x, y > 0)
Solving x+y = 6 and x-y = 2, we get x = 4 and y = 2 (Answer)

Example 3: Divide (a³ + b³ + c³ – 3abc) by (a + b + c) and determine the quotient.

Solution: a³ + b³ + c³ – 3abc

= (a+b+c)(a² + b² + c² - ab - ac - bc)

Quotient = [(a+b+c)(a² + b² + c² - ab - ac - bc)] ÷ (a+b+c)

= a² + b² + c² - ab - ac - bc

Magnitude of the quotient is 2. (Answer)

Example 4: Find the successive product of (x + y), (x – y), and (x² + y²).

Solution: Serial product = (x + y)(x – y)(x² + y²)
= (x² - y²)(x² + y²)
= (x²)² – (y²)²
= x⁴ – y⁴ (Answer)

Some Questions of Algebra Formula

1. If x+y = 3 and xy = 2, what is (x – y)²?
2. If a+b = 8 and ab = 15, what are the values of a and b?
3. If a+b = 5 and ab = 6, find the value of a² – b².
4. If x = 29 and y = 14, what is the value of (4x² + 9y² + 12xy)?

All Algebraic Identities

Algebraic identities are essential equations true for all values of the variables involved. Here are more crucial identities.

1. Basic Algebraic Identities

• Square of a Sum:
  $(a + b)^2 = a^2 + 2ab + b^2$ 
• Square of a Difference:
  $(a – b)^2 = a^2 – 2ab + b^2$ 
• Product of a Sum and Difference:
  $(a + b)(a – b) = a^2 – b^2$ 
• Cubic of a Sum:
  $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$ 
• Cubic of a Difference:
  $(a – b)^3 = a^3 – 3a^2b + 3ab^2 – b^3$ 
• Sum of Cubes:
  $a^3 + b^3 = (a + b)(a^2 – ab + b^2)$ 
• Difference of Cubes:
  $a^3 – b^3 = (a – b)(a^2 + ab + b^2)$ 

2. Multinomial Algebraic Identities

• Square of a Trinomial:
  $(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$ 
• Cubic of a Trinomial:
  $(a + b + c)^3 = a^3 + b^3 + c^3 + 3(a + b)(b + c)(c + a)$ 

3. Special Algebraic Identities

• General Binomial Theorem:
  $(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k$ 
• (a + b + c)² Expanded:
  $(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$ 
• Sum of Powers of Roots:
  $a^n – b^n = (a – b)(a^{n-1} + a^{n-2}b + \cdots + b^{n-1})$ 

4. Quadratic Algebraic Identities

• For any quadratic polynomial:
  $ax^2 + bx + c = a(x – \alpha)(x – \beta)$where

  $\alpha$ and b are the roots of the equation

  $ax^2 + bx + c = 0$

5. Important Algebraic Products

• Sum of squares of two numbers:
  $a^2 + b^2 = \frac{(a + b)^2 + (a – b)^2}{2}$ 
• Product of four numbers:
  $(a^2 + b^2)(c^2 + d^2) = (ac + bd)^2 + (ad – bc)^2$ 

6. Miscellaneous Algebraic Identities

• Lagrange’s Identity:
  $(a^2 + b^2)(c^2 + d^2) = (ac + bd)^2 + (ad – bc)^2$ 
• Expression for
  $(a + b + c)^3$ 

  $(a + b + c)^3 = a^3 + b^3 + c^3 + 3(a + b)(b + c)(c + a)$

These identities are fundamental for simplifying expressions and mastering algebraic manipulation. Download our Algebra Formulas PDF for further study.

Check out the complete Algebraic Formulas PDF for Class 10 students here: Math Algebra Formulas PDF

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