Algebra Formulas

Algebra Formulas are the basic formulas that are used to simplify algebraic expressions. Algebraic Formulas form the basis for solving various complex problems. Algebraic Formulas help solve algebraic equations, quadratic equations, polynomials, trigonometry equations, probability questions, and others.

An identity is a true equation for any values assigned to the variables. Algebraic Identities are used to solve various equations. For Algebraic Identities, L.H.S is always equal to R.H.S.

Algebraic formulas are equations that require algebraic expression on both sides of “equal to” i.e. both on LHS and RHS. Algebraic formulas are generally true for all the values. Algebraic Formula simplifies algebraic equations and is required to solve various problems in mathematics. Algebraic formulas for various classes are discussed below in this article.

Some important algebraic identities are:

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

For class 9 Logarithms 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

Exponents are ways to represent higher power. Laws of exponent 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

Read More about Law of Exponent.

“Quadratic Formula” is an important algebraic formula 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 variable a, b is coefficient 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 Sequence

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 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,

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)!]

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

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
  where θ is the angle between the vectors a and 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 are given by [a b c ] = a ⋅ (b × c) = (a × b) ⋅ c.

Also, Check

• Algebraic Identities of Polynomials
• Vector Algebra
• Percentage Formula

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

Solution:

Using the algebraic formula,

(a + b)² = a² + b² + 2ab

(2x + 3)² = (2x)² + 3² + 2 × 2x × 3

(2x + 3)² = 4x² + 9 + 12x

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

Solution:

Using the algebraic formula,

(a – b)² = a² + b² – 2ab

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

(5 – 3)² = 25x² + 9y² – 30xy

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

Solution:

Using the algebraic formula,

(a + b)(a – b) = a² – b²

105×95 = (100+5)(100-5)
             = 100² – 5²
             = 10000 – 25
             = 9975

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

Solution:

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

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)

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

Solve for y if (4y − 5)² = 121

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

Solve for c if (c+3)³ = 512

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

What is the Formula for a² – b² in Algebra?

The formula for a²– b² defined in algebra is 

a²– b² = (a+b)(a-b)

This formula is also called the difference of squares formula.

What are Algebraic Expressions?

Algebraic expressions are combinations of variables and constants using arithmetic operations such as Addition, Subtraction, Multiplication, and Division. 

Example: 11 x³ + 5x is an algebraic expression. This expression has two terms 11 x³ and 5x.

What are the Three Basic Algebra Formulas?

The three basic formulas of Algebra are:

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

Write the Simplified Form of (a+b)².

(a+b)² can be written in simplified form as  (a²+ 2ab + b²)