BrightChamps Logo
Login
Creative Math Ideas Image
Live Math Learners Count Icon106 Learners

Last updated on July 4th, 2025

Math Whiteboard Illustration

Algebra Formulas

Professor Greenline Explaining Math Concepts

In mathematics, algebra formulas are important as they form the foundation for polynomials, calculus, trigonometry, and quadratic equations. These formulas help solve and simplify algebraic expressions. In this article, algebraic formulas will be discussed in detail.

Algebra Formulas for Vietnamese Students
Professor Greenline from BrightChamps

What are Algebra Formulas?

Algebra formulas are rules or equations that help with factoring, expanding, and simplifying expressions. We can use these formulas to solve complex algebraic equations efficiently. Here are some algebraic formulas: 

 

  • (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
Professor Greenline from BrightChamps

Exponent Rules/Laws:

To solve the expressions involving powers or exponents, we use the exponent rules. These rules are used to simplify expressions with powers. The exponent rules are: 

 

Rule Formula
Product Rule am × an = am + n
Quotient Rule am ÷ an = am - n
Power of a Power Rule (am)n = amn
Power of a Product Rule (ab)m = ambm
Power of a Quotient Rule (a/b)m = (am)/(bm)
Zero Exponent Rule a0 = 1
Negative Exponent Rule a-m = 1/am

 

Professor Greenline from BrightChamps

Properties of Logarithms:

Logarithms are used to solve multiplication and division of numbers with powers in simple ways. This makes them an effective tool to work with algebraic formulas with exponents. The relationship between the exponent and logarithm is: xm = a ⇒ logx a = m

 

Some commonly used log algebraic formulas are: 

 

loga a = 1

loga 1 = 0

loga (xy) = loga x+ loga y

loga (x/y) = loga x - loga

loga (xm) = m loga x

loga x = logcx/logc

a logax = x
 

Professor Greenline from BrightChamps

Quadratic Formula:

The quadratic formula is one of the two methods to solve a quadratic equation. The standard form of a quadratic equation is ax2 + bx + c = 0. The value of the variable x can be found by using the formula:     
x = -b ± √b2 - 4ac/2a

Professor Greenline from BrightChamps

Permutations and Combination Formulas:

In algebra, permutations and combinations are formulas that help us identify the number of ways something can be arranged. Permutations refer to arrangements of items where the order matters, and combinations are the selection of items where order does not matter. 

 

Factorial formula: n! = n × (n - 1) × (n - 2) × …. × 3 × 2 × 1
 
Permutations formula: nPr = n! / (n - r)!

Combination formula: nCr = n! / (r! (n - r)!)

Binomial theorem: (x + y)n =  nC0xny0 + nC1xn-1y1 + nC2xn-2y2 + …. + nCn-1xyn-1 + nCnx0yn

Professor Greenline from BrightChamps

Vector Algebra Formula:

The vector algebra formula is used to solve problems related to directions and magnitude. Some important vector formulas are: 

 

For any three vectors a, b, and c in a 3D space

 

  • The magnitude of a = xi + yj + zk, 
    So, |a| = √x2 + y2 + z2
     
  • The unit vector along a = a/|a|
     
  • Dot product: ab = |a| |b| cosθ, where θ is the angle between two vectors a and b
     
  • Cross product: a × b = absinθn̂
     
  • Scalar triple product: [a b c] = a  (b × c) = (a × b) · c
Professor Greenline from BrightChamps

What are the Formulas for Algebraic Identities?

Algebraic identities are the equations that hold true for all values of the variables involved. It means LHS = RHS of the equation. Some common algebraic identities are - 

 

  •  (a + b)2 = a2 + 2ab + b2
     
  •  (a - b)2 = a2 - 2ab + b2
     
  • (a + b)(a - b) = a2 - b2
     
  • (x + a) (x + b) = x2 + (a + b)x + ab
Professor Greenline from BrightChamps

What are Algebra Formulas of Functions?

The algebraic function expresses a relationship between two variables; it is written in the form y = f(x). Where x is the input and y is the output. For example, if x = 4, then f(x) = f(4) = 42 = 16. 

Professor Greenline from BrightChamps

What are Algebra Formulas of Fractions?

In algebra, fractions that contain variables are called rational expressions. We can add, subtract, multiply, and divide fractions. The algebraic expression of fractions is:

 

  • x /y + z/w = (xw + yz) / (yw)
     
  • x /y - z/w = (xw - yz) / (yw)
     
  • x /y × z/w = xz / yw
     
  • x /y ÷ z/w = xw / yz
Professor Greenline from BrightChamps

Real-Life Applications of Algebra Formulas

In real life, we use algebraic formulas from managing personal finances and cooking to understanding scientific concepts and designing technology. Here are a few applications of algebraic formulas:

 

  • To create budgets and track expenses, we can set up an equation to balance income and spending.

 

  • To calculate the loan payment and compound interest, there are algebraic formulas.

 

  • In cooking and baking, to scale recipes up or down, we can use algebra.  

 

  • In health and fitness, to calculate calorie counting and exercise regimens, we can use algebraic formulas.
Max Pointing Out Common Math Mistakes

Common Mistakes and How to Avoid Them in Algebra Formulas

Students often think that algebra formulas are difficult and make mistakes. Here are some mistakes that students make and repeat, but by learning these mistakes, students can master algebra formulas.

Mistake 1

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Incorrect expansion of (a + b)2 and (a - b)2

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Students often think that (a + b)² = a² + b² and (a - b)² = a² - b² and ignore 2ab which is wrong. So students should remember that, (a + b)² = a² + 2ab + b² and (a - b)² = a² -2ab + b².

Mistake 2

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Adding unlike terms

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Sometimes students add the terms with different variables or exponents, for example, x2 + x = x3, which is wrong, as both the addends have different exponents. Students should always remember that we can add only like terms, that is, the terms with the same variables raised to the same power.

Mistake 3

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Mixing identities

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Many students assume that the expansion of (a + b)2 is the same as the expansion of (a + b)3. In other words, they think that (a + b)3 = a2 + 2ab + b2, which is wrong. Always remember that the expansion is different for (a + b)² and (a + b)³. So memorize the formulas, (a + b)² = a2 + 2ab + b2 and (a + b)3 = a3 + 3a2b + 3ab2 + b3.

Mistake 4

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Sign errors when working with negative signs

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Sign errors are common among students, especially when dealing with negative signs.

 

For instance, students think -(2x + 1) = -2x + 1 instead of -2x -1. It happens because while using signs with parentheses, students often make silly mistakes. It is always good to keep this small detail in mind.

Mistake 5

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Confusing variables with constants

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Mixing up constants and variables is common among students. In the equation y = 2x + 3, x is the variable, and the numbers 2 and 3 are constants. Practice identifying constants and variables in other equations to avoid this mistake.

arrow-right
Max from BrightChamps Saying "Hey"

Solved Examples of Algebra Formulas

Ray, the Character from BrightChamps Explaining Math Concepts
Max, the Girl Character from BrightChamps

Problem 1

Using algebra formulas, find (x +7)²?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

(x + 7)2 = x2 + 14x + 49

Explanation

To find the value of (x + 7)², we use the algebraic identity (a + b)2 = a2 + 2ab + b2

Here, a = x and b = 7

So, (x + 7)² = x² + (2 × x × 7) + 7²

= x² + 14x + 49

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 2

Find the value of (x + 2)(x + 8), using algebraic identity.

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

(x + 2) (x + 8) = x² + 10x + 16

Explanation

The value of (x + 2) (x + 8) is calculated by using the identity (x + a) (x + b) = x2 + x(a + b) + ab

Here, a = 2 and b = 8

So, (x + 2) (x + 8) = x2 + x(a + b) + ab

= x² + x(2 + 8) + (2 × 8)

= x² + 10x + 16

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 3

Apply the identity (a + b) (a - b) = a² - b² to evaluate 102² - 98²

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

The value of 102² - 98² = 800

Explanation

The value of 102² - 98² is calculated using the identity:
 
(a + b) (a - b) = a² - b², where a = 102 and b = 98

So, 1022 - 982 = (102 + 98) (102 - 98)

= 200 × 4 = 800

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 4

Solve the quadratic equation x² - 7x + 12 = 0 using quadratic formula

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

The value of x = 3 or x = 4

Explanation

The quadratic formula is: x = -b ± √b2 - 4ac/2a

For x2 + bx + c = 0, here a = 1, b = -7, c = 12

x = -(-7) ± √(-7)2 - 4 × 1 × 12/2 × 1

= 7 ± √49 - 48/2  = (7 ± 1) / 2

So, x = (7 + 1)/2 = 8/2 = 4

x = (7 - 1)/2 = 6/2 = 3

Therefore, the value of x can be 4 or 3

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 5

Find the product of (x - 3)(x + 3) using algebraic formulas.

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

The value of (x - 3)(x + 3) = x2 -9

Explanation

The value of (x - 3) (x + 3) is calculated by using the identity (a - b) (a + b) = a2 - b²

Here, a = x and b = 3

(a - b) (a + b) = x² - 3²

= x² - 9

Max from BrightChamps Praising Clear Math Explanations
Ray Thinking Deeply About Math Problems

FAQs on Algebra Formulas

1.What are algebraic formulas?

Math FAQ Answers Dropdown Arrow

2.What is the quadratic formula?

Math FAQ Answers Dropdown Arrow

3.What is the basic algebraic formula?

Math FAQ Answers Dropdown Arrow

4.What are the formulas for an arithmetic sequence?

Math FAQ Answers Dropdown Arrow

5.How can students remember the algebraic formulas?

Math FAQ Answers Dropdown Arrow

6.How does learning Algebra help students in Vietnam make better decisions in daily life?

Math FAQ Answers Dropdown Arrow

7.How can cultural or local activities in Vietnam support learning Algebra topics such as Algebra Formulas ?

Math FAQ Answers Dropdown Arrow

8.How do technology and digital tools in Vietnam support learning Algebra and Algebra Formulas ?

Math FAQ Answers Dropdown Arrow

9.Does learning Algebra support future career opportunities for students in Vietnam?

Math FAQ Answers Dropdown Arrow
Math Teacher Background Image
Math Teacher Image

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Max, the Girl Character from BrightChamps

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.

INDONESIA - Axa Tower 45th floor, JL prof. Dr Satrio Kav. 18, Kel. Karet Kuningan, Kec. Setiabudi, Kota Adm. Jakarta Selatan, Prov. DKI Jakarta
INDIA - H.No. 8-2-699/1, SyNo. 346, Rd No. 12, Banjara Hills, Hyderabad, Telangana - 500034
SINGAPORE - 60 Paya Lebar Road #05-16, Paya Lebar Square, Singapore (409051)
USA - 251, Little Falls Drive, Wilmington, Delaware 19808
VIETNAM (Office 1) - Hung Vuong Building, 670 Ba Thang Hai, ward 14, district 10, Ho Chi Minh City
VIETNAM (Office 2) - 143 Nguyễn Thị Thập, Khu đô thị Him Lam, Quận 7, Thành phố Hồ Chí Minh 700000, Vietnam
Dubai - BrightChamps, 8W building 5th Floor, DAFZ, Dubai, United Arab Emirates
UK - Ground floor, Redwood House, Brotherswood Court, Almondsbury Business Park, Bristol, BS32 4QW, United Kingdom