Algebra Formulas

Algebra Formulas are standard mathematical equations used in the branch of algebra to solve various questions based on the concepts of algebra. This article provides compilation of all the algebra formulas that come handy when revising at the last moment for the exam. This article also provides algebra formulas for students for each class which are mentioned in their curriculum.

What are Algebra Formulas?

Algebra formulas are mathematical expressions or equations that represent relationships between variables or quantities. In algebra, letters (variables) are used to represent unknown quantities, and algebraic formulas describe how these variables relate to each other. An algebra formula consists of the following elements:

Variables: These are symbols (usually letters) that represent quantities that can vary or change.

Constants: These are fixed values that do not change.

Operations: Algebraic formulas often involve arithmetic operations such as addition, subtraction, multiplication, division, exponentiation, and roots.

Equations and Inequalities: Algebraic formulas can be expressed as equations (where two expressions are equal) or inequalities (where one expression is greater than or less than another).

Algebraic Identities

Some important algebraic identities are

Algebraic Properties

Various algebraic properties are mentioned below:

Commutative Property

• Addition: a+b = b+a
• Multiplication: ab = ba

Associative Property

• Addition: (a+b)+c = a+(b+c)
• Multiplication: (ab)c = a(bc)

Distributive Property

• a(b+c)=ab+ac
• a(b–c)=ab–ac

Identity Element

• Addition: a+0=a
• Multiplication: a×1=a

Inverse Element:

• Addition: a+(−a)=0
• Multiplication: a×1/a=1, where, a≠0

Basic Formulas in Algebra

The section contains all basic algebra formulas to solve the fundamental and complicated mathematical problems for secondary standard students. 

• 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)
• If n is a natural number aⁿ – bⁿ = (a – b)(a^n-1 + a^n-2b+…+ b^n-2a + b^n-1)
• If n is even (n = 2k), aⁿ + bⁿ = (a + b)(a^n-1 – a^n-2b +…+ b^n-2a – b^n-1)
• If n is odd (n = 2k + 1), aⁿ + bⁿ = (a + b)(a^n-1 – a^n-2b +a^n-3b²…- b^n-2a + b^n-1)
• (a + b + c + …)² = a² + b² + c² + … + 2(ab + ac + bc + ….)
• Laws of Exponents (a^m)(aⁿ) = a^m+n ; (ab)^m = a^mb^m ; (a^m)ⁿ = a^mn

Algebra Formulas for Class 8

This article discusses algebra formulae for class 8. The algebraic formulae for three variables a, b, and c are as follows:

• (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

The algebra formula for class 9 are mentioned below

•  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
• (x³ + y³ + z³ – 3xyz) = (x + y + z)(x² + y² + z² – xy – yz – zx)
• (x + y + z) = 0 then (x³ + y³ + z³) = 3xyz

Exponent Formulas

The exponent rules in algebra are mentioned below:

• (a^m x aⁿ) = a^{(m + n)}
• (a^m/aⁿ) = a^(m-n)
• a⁰ = 1
• a^-n = 1/aⁿ
• a^m x b^m = (a x b)^m
• {(a)^m}ⁿ = a^mn

Logarithm Formulas

The logarithm formulas in algebra are mentioned below:

• loga (xy) = log_a x + log_a y
• loga (x/y) = log_a x – log_a y
• loga x^m = m log_a x
• log_aa = 1
• loga 1 = 0

Algebra Formula Class 10

The algebra formula for class 10 are discussed below:

For a polynomial ax² + bx + c, if roots are p and q

Then,

• Sum of Roots i.e. p + q = -b/a
• Product of roots i.e. pq = c/a

For a polynomial ax³ + bx² + cx + d, if roots are p, q and r

• Sum of roots i.e. p + q + r = -b/a
• Sum of products of roots taken two at a time i.e. pq + qr + pr = c/a
• Product of roots i.e. pqr = -d/a

Quadratic Equation Formula

For a quadratic equation given as ax² + bx + c, its roots are given as

X = (-b ± √D)/2a

Where, D is discriminant and D = b² – 4ac

AP Formulas

For a sequence given as a, a+d, a+2d, ……, a + (n – 1)d, ….

Common Difference: d = (a₂ – a₁), where a2 and a1 are successive terms and preceding terms respectively.

• General Term (nth term): an = a + (n – 1)d
• nth Term from the last term: an = l – (n – 1)d
• Sum of first n terms: S_n = n/2[2a + (n – 1)d]

Algebra Formula Class 11

Algebra formulas for class 11 is given as below:

Inequality Formulas

• Addition Property of Inequality
  • (i) If x > y, then (x + z) > (y + z)
  • (ii) If x < y, then (x + z) < (y + z)

• Subtraction Property of Inequality
  • (i) If x > y, then (x − z) > (y − z)
  • (ii) If x < y, then (x − z) < (y − z)
• Multiplication Property of Inequality:
  • (i) If x > y and z > 0, then xz > yz
  • (ii) If x< y and z > 0, then xz < yz
  • (iii) If x > y and z < 0, then xz < yz
  • (iv) If x < y and z < 0, then xz > yz
• Division Property of Inequality:
  • (i) If x > y and z > 0, then x/z > y/z
  • (ii) If x < y and z > 0, then x/z < y/z
  • (iii) If x > y and z < 0, then x/z < y/z
  • (iv) If x < y and z < 0, then x/z > y/z

Permutation and Combination Formulas

• P(n, r) = n!/(n − r)!
• C(n, r) = n!/(n − r)!r!

Binomial Theorem

(a + b)ⁿ = aⁿ + (ⁿC₁)a^n-1b + (ⁿC₂)a^n-2b² + … + (ⁿC_n-1)ab^n-1 + bⁿ

GP Formulas

For a GP given as Sequence: a, ar, ar², …., ar^(n-1), …

Common Ratio: r = ar^(n-1)/ar^(n-2), where ar^(n-1) and ar^(n-2) are successive term and preceding term respectively.

• General Term (nth term): a_n = ar^(n-1)
• nth Term from the last term: a_n = 1/r^(n-1)
• Sum of first n terms: S_n = a(1 – rⁿ)/(1 – r) if r < 1; S_n = a(rⁿ -1)/(r – 1) if r > 1

Algebra Formula Class 12

The algebra formulas for class 12 are mentioned below:

Addition of Matrices
If \( \mathbf{A} \) and \( \mathbf{B} \) are two matrices of the same dimensions \( m \times n \), their sum \( \mathbf{C} \) is given by:
\[ \mathbf{C} = \mathbf{A} + \mathbf{B} \]
\[ c_{ij} = a_{ij} + b_{ij} \quad \text{for all } i \text{ and } j \]

Multiplication of Matrices
If \( \mathbf{A} \) is an \( m \times n \) matrix and \( \mathbf{B} \) is an \( n \times p \) matrix, their product \( \mathbf{C} \) is an \( m \times p \) matrix given by:
\[ \mathbf{C} = \mathbf{A} \mathbf{B} \]
\[ c_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj} \quad \text{for all } i \text{ and } j \]

Scalar Multiplication of Matrix
If \( \mathbf{A} \) is a matrix and \( k \) is a scalar, the product \( k\mathbf{A} \) is obtained by multiplying each element of \( \mathbf{A} \) by \( k \):
\[ (k\mathbf{A})_{ij} = k \cdot a_{ij} \quad \text{for all } i \text{ and } j \]

Determinant of a Matrix
For a \( 2 \times 2 \) matrix \( \mathbf{A} \):
\[ \mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]
The determinant is given by:
\[ \det(\mathbf{A}) = ad – bc \]

For a \( 3 \times 3 \) matrix \( \mathbf{A} \):
\[ \mathbf{A} = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]
The determinant is given by:
\[ \det(\mathbf{A}) = a(ei – fh) – b(di – fg) + c(dh – eg) \]

Transpose of a Matrix
If \( \mathbf{A} \) is an \( m \times n \) matrix, its transpose \( \mathbf{A}^T \) is an \( n \times m \) matrix given by:
\[ (\mathbf{A}^T)_{ij} = a_{ji} \quad \text{for all } i \text{ and } j \]

Adjoint of a Matrix
The adjoint (or adjugate) of a matrix \( \mathbf{A} \) is the transpose of the cofactor matrix of \( \mathbf{A} \):
\[ \text{adj}(\mathbf{A}) = \mathbf{C}^T \]
where \( \mathbf{C} \) is the cofactor matrix.

Inverse of a Matrix
For a \( 2 \times 2 \) matrix \( \mathbf{A} \):
\[ \mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]
If \( \det(\mathbf{A}) \neq 0 \), the inverse \( \mathbf{A}^{-1} \) is given by:
\[ \mathbf{A}^{-1} = \frac{1}{\det(\mathbf{A})} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]

For a \( 3 \times 3 \) matrix \( \mathbf{A} \):
\[ \mathbf{A}^{-1} = \frac{1}{\det(\mathbf{A})} \text{adj}(\mathbf{A}) \]
provided that \( \det(\mathbf{A}) \neq 0 \).

Unit Vector
A unit vector is a vector with a magnitude of 1. To find the unit vector \(\mathbf{u}\) in the direction of a given vector \(\mathbf{v}\), you use the formula:
\[ \mathbf{u} = \frac{\mathbf{v}}{|\mathbf{v}|} \]
where \( |\mathbf{v}| \) is the magnitude of \(\mathbf{v}\).

Magnitude of a Vector
For a vector \(\mathbf{v} = \langle v_1, v_2, \ldots, v_n \rangle \) in \( n \)-dimensional space, the magnitude \( |\mathbf{v}| \) is given by:
\[ |\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2} \]

For a 3-dimensional vector \(\mathbf{v} = \langle v_x, v_y, v_z \rangle\):
\[ |\mathbf{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2} \]

Dot Product of Vectors
For two vectors \(\mathbf{a} = \langle a_1, a_2, \ldots, a_n \rangle \) and \(\mathbf{b} = \langle b_1, b_2, \ldots, b_n \rangle \) in \( n \)-dimensional space, the dot product \( \mathbf{a} \cdot \mathbf{b} \) is given by:
\[ \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + \cdots + a_n b_n \]

For 3-dimensional vectors \(\mathbf{a} = \langle a_x, a_y, a_z \rangle\) and \(\mathbf{b} = \langle b_x, b_y, b_z \rangle\):
\[ \mathbf{a} \cdot \mathbf{b} = a_x b_x + a_y b_y + a_z b_z \]

Cross Product of Vectors
For two vectors \(\mathbf{a} = \langle a_x, a_y, a_z \rangle\) and \(\mathbf{b} = \langle b_x, b_y, b_z \rangle\) in 3-dimensional space, the cross product \( \mathbf{a} \times \mathbf{b} \) is given by:
\[ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix} \]
Where \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) are the unit vectors in the x, y, and z directions, respectively.

Expanding the determinant, we get:
\[ \mathbf{a} \times \mathbf{b} = \left( a_y b_z – a_z b_y \right) \mathbf{i} – \left( a_x b_z – a_z b_x \right) \mathbf{j} + \left( a_x b_y – a_y b_x \right) \mathbf{k} \]

In vector notation, this can be written as:
\[ \mathbf{a} \times \mathbf{b} = \langle a_y b_z – a_z b_y, a_z b_x – a_x b_z, a_x b_y – a_y b_x \rangle \]

Algebra Formula Solved Examples

Example 1: Find out the value of the term, (5x + 4)² using algebraic formulas.

Solution:

Using the algebraic formula,

(a + b)² = a² + b² + 2ab

(5x + 4)² = (5x)² + ⁴² + 2 × 5x × 4

(5x + 4)² = 25x² + 16 + 40x

Example 2: Find out the value of the term, (9x – 5y)² using algebraic formulas.

Solution:

Using the algebraic formula,

(9x – 5y)² = (9x)² + (5y)² – 2 × 9x × 5y

(9x – 5y)² = 9x² + 25y² – 90xy

Example 3: Find out the value of, 205×195 using algebraic formulas.

Solution:

Using the algebraic formula,

(a + b)(a – b) = a² – b²

205×195 = (200+5)(200-5)
= 200² – 5²
= 40000 – 25
= 39975