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.
Table of Content
- What are Algebra Formulas?
- Algebraic Identities
- Algebra Formulas for Class 8
- Algebra Formulas for Class 9
- Law of Exponent
- Algebra Formulas for Class 10
- Formulas for Arithmetic Sequence
- Formulas for Geometric Sequence
- Algebra Formulas for Class 11
- Algebra Formulas for Class 12
- Solved Examples on Algebra Formulas
- Summarizing Algebra Formulas
- Practice Problems on Algebra Formulas
What are Algebra Formulas?
Algebraic formulas are equations that require algebraic expression on both sides of βequal toβ 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.
Algebraic Identities
Some important algebraic identities are:
(a + b)2 | a2 + b2 + 2ab |
(a β b)2 | a2 + b2 β 2ab |
(a + b)(a β b) | a2 β b2 |
(x + a)(x + b) | x2 + x(a + b) + ab |
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 β b)2 = a2 β 2ab + b2
- (a + b)(a β b) = a2 β b2
- (a + b)3 = a3 + 3a2b + 3ab2 + b3
- (a β b)3 = a3 β 3a2b + 3ab2 β b3
- a3 + b3 = (a + b)(a2 β ab + b2)
- a3 β b3 = (a β b)(a2 + ab + b2)
- (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
Algebra Formulas for Class 9
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:
- loga (xy) = loga x + loga y
- loga (x/y) = loga x β loga y
- loga xm = m loga x
- loga a = 1
- loga 1 = 0
Law of Exponent
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.
- amΓ an = am + n
- am/an = am β n
- (am)n = amn
- (ab)m = amΓ bm
- a0 = 1
- a-m = 1/am
Read More about Law of Exponent.
Algebra Formulas for Class 10
βQuadratic Formulaβ 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, β¦}
- nth term, an = a + (n β 1) d
- Sum of the first n terms, Sn = n/2 [2a + (n β 1) d]
Formulas for Geometric Sequence
For any given geometric sequence {a, ar, ar2, β¦}
- nth term, an = a rn β 1
- Sum of the first n terms, Sn = a (1 β rn) / (1 β r)
- Sum of infinite terms when r<1, S = a / (1 β 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,
- n! = n Γ (n β 1) Γ (n β 2) Γ β¦ Γ 3 Γ 2 Γ 1
- nPr = n! / (n β r)!
- nCr = n!/[r!(nβr)!]
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| =β(x2+y2+z2).
- Unit vector along a is a / |a|
- Dot product of two vectors a and b is defined as a β
b = |a| |b| cos ΞΈ
where ΞΈ is the angle between the vectors a and b. - Cross product of vectors a and b is defined as a Γ b = |a| |b| sin ΞΈ
where ΞΈ 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 β (b Γ c) = (a Γ b) β c.
Also, Check
Solved Examples on Algebra Formulas
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 Γ 2x Γ 3
(2x + 3)2 = 4x2 + 9 + 12x
Example 2: Find out the value of the term, (5x β 3y)2 using algebraic formulae.
Solution:
Using the algebraic formula,
(a β b)2 = a2 + b2 β 2ab
(5x β 3y)2 = (5x)2 + (3y)2 β 2 Γ 5x Γ 3y
(5 β 3)2 = 25x2 + 9y2 β 30xy
Example 3: Find out the value of, 105Γ95 using algebraic formulae.
Solution:
Using the algebraic formula,
(a + b)(a β b) = a2 β b2
105Γ95 = (100+5)(100-5)
= 1002 β 52
= 10000 β 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 = [βb Β± β(b2 β 4ac)] / 2a
= [β6 Β± β(62 β 4(1)(8))] / 2(1)
= [β6 Β± β(36 β 4(1)(8))] / 2
= [β6 Β± β(36 β 32)] / 2
= [β6 Β± β4] / 2
= (-6 + 2) / 2 and (-6 β 2) / 2
= -4/2 and -8/2
= -2 and -4Thus, the values of x are -2 and -4
Summarizing Algebra Formulas
All the important algebra formulas are
- (a + b)2 = a2 + 2ab + b2
- (a β b)2 = a2 β 2ab + b2
- (a + b)(a β b) = a2 β b2
- (a + b)3 = a3 + 3a2b + 3ab2 + b3
- (a β b)3 = a3 β 3a2b + 3ab2 β b3
- a3 + b3 = (a + b)(a2 β ab + b2)
- a3 β b3 = (a β b)(a2 + ab + b2)
- (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
- loga (xy) = loga x + loga y
- loga (x/y) = loga x β loga y
- loga xm = m loga x
- loga a = 1
- loga 1 = 0
- amΓ an = am + n
- am/an = am β n
- (am)n = amn
- (ab)m = amΓ bm
- a0 = 1
- a-m = 1/am
- nth term of a AP, an = a + (n β 1) d
- Sum of the first n terms, Sn = n/2 [2a + (n β 1) d]
- nth term of a GP, an = a rn β 1
- Sum of the first n terms, Sn = a (1 β rn) / (1 β r)
- Sum of infinite terms when r < 1, S = a / (1 β r)
Practice Problems on Algebra Formulas
Find the value of x if (2x + 3)2 = 49.
Solve for y if (4y β 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 logβ‘7(a3) = 6 and logβ‘7(a) = 2.
FAQs on Algebra Formulas
What is the Formula for a2 β b2 in Algebra?
The formula for a2β b2 defined in algebra is
a2β 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 β b)2 = a2 + b2 β 2ab
- (a + b)(a β b) = a2 β b2
Write the Simplified Form of (a+b)Β².
(a+b)Β² can be written in simplified form as (aΒ²+ 2ab + bΒ²)