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:
| (a + b)2 | a2 + b2 + 2ab |
| (a - b)2 | a2 + b2 - 2ab |
| (a + b)(a - b) | a2 - b2 |
| (x + a)(x + b) | x2 + x(a + b) + ab |
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)(an-1 + an-2b+…+ bn-2a + bn-1)
- (am)(an) = am+n ; (ab)m = ambm ; (am)n = amn
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:
- loga (xy) = loga x + loga y
- loga (x/y) = loga x - loga y
- loga xm = m loga x
- loga a = 1
- loga 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, an = a + (n - 1) d
- Sum of the first n terms, Sn = n/2 [2a + (n - 1) d]
Formulas for Geometric Sequences
For any given geometric sequence {a, ar, ar2, ...}
- nth term, an = a rn - 1
- Sum of the first n terms, Sn = 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)(an-1 + an-2b +an-3b2 + …+ abn-2 + bn-1)
If n is odd, then
- an + bn = (a + b)(an-1 – an-2b +an-3b2 – …- abn-2 + bn-1)
- (am )(an ) = am+n
- (ab)m = ambm
- (am)n = amn
- n! = n × (n - 1) × (n - 2) × ... × 3 × 2 × 1
- nPr = n! / (n-r)!
- nCr = 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
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.
- am× an = am + n
- am/an = am - n
- (am)n = amn
- (ab)m = am× bm
- a0 = 1
- a-m = 1/am
Solved Question
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
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 log7(a3) = 6 and log7(a) = 2.
Question 6. Simplify the expression using algebraic identities:
Question 7. Factorize the expression:
Question 8. Expand and simplify:
Question 9. Simplify the expression using logarithmic properties:
Question 10. Determine the 5th term of the geometric sequence where the first term