Algebra Formula: Important Formulas in Algebra & Solved Examples

Alphabetical notation in any equation of mathematics and science is called algebra formula. It consists of symbols that are known as variables that represent quantity without a fixed value. The most common letters used to express algebraic problems and equations are X, Y, A, and B. Just as an engineer's life mainly revolves around wires, gadgets, current and electricity, similarly for a mathematician, algebra is an important tool, their life also revolves around the same.

Real numbers, complex numbers, matrices, vectors, and other concepts are all part of algebra.​ Algebra formulas have a wide range of applications, which is why it is considered one of the important units. In this article, we will cover all the important algebraic formulas which are used in all classes starting from class 8.

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Algebraic Formulas

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Inclusion of new topics in Mathematics is usually seen after class 8, depending on the type of board (CBSE or state or ICSE) and the books run by different schools. Before this, students may be aware of different types of numbers like real numbers, prime numbers etc. but after class 8, they are usually familiar with an important chapter “Algebra” which forms the basis of complex chapters like Trigonometry.

Algebra is a branch of mathematics that deals with both numbers and letters. The value of numbers is fixed and the letters or alphabets represent the unknown quantities in the algebraic formula. A combination of numbers, letters, factorials, matrices, etc. is used to form an algebraic equation or algebraic formula. 

Here is the list of all the important algebraic formulas: 

1. (a+b)² = a² + 2ab + b²
2. (a-b)2 = a² – 2ab + b²
3. (a+b)(a-b) = a² – b²
4. (x +a)(x+b) = x² +(a+b)x +ab
5. (x+a)(x-b) = x² + (a-b)x – ab
6. (x-a)(x+b) = x² + (b-a)x – ab
7. (x-a)(x-b) = x² – (a+b)x +ab
8. (a+b)3 = a³ + b³ + 3ab(a+b)
9. (a-b)3 = a³ – b³ – 3ab (a-b)
10. (x+y+z)² = x² + y² +z² + 2xy + 2yx + 2xz
11. (x+y-z)² = x² + y² +z² + 2xy – 2yz – 2xz
12. (x – y +z)² = x² + y² +z²  - 2xy – 2yz +2xz 
13. (x-y-z)² = x² + y² +z²  - 2xy +2yz -2xz
14. X³ + y³ +z³ – 3xyz = (x+y+z) (x2+ y2 +z2)
15. (x+a)(a+b)(a+c) = x³ + (a+b+c) x² + (ab +bc+ ca)x + abc
16. X³ + y³ = (x+y) (x² – xy + y²)
17. X³ – y³ = (x- y) (x² + xy + y²)
18. X² + y² + z² –xy – yz – zx = 12[(x-y)² + (y-z)² + (z-x)²
19. X² + y² = ½[(x +y)² + (x – y)² ]
20. a1/x = xa
21. (a/b)^x = a²/b²
22. (ab)^x = a^xb^x
23. (a^m)ⁿ = a^mn
24. a^m/aⁿ = a^m-n
25. Laws of Exponents a^m x aⁿ = a^m+n
26. a⁸- b⁸ = (a⁴ + b⁴) (a² +b²) (a+b)(a-b)
27. a⁴ + a² + 1 = (a² + a +1)(a²-a + 1)
28. a⁴ + a²b² + b⁴ = (a² + ab + b² )(a² –ab + b²)

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Linear Equation in Two Variables

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Algebra formulas are classified into two parts: basic and advanced algebra. Basic algebraic formulas are generally used by students up to class 10. Next, advanced algebraic equations are used to solve complex problems and write equations. Application of advanced level algebraic formulas is also seen in the engineering field.

An equation is said to be a linear equation in two variables if it is written in the form of ax + by + c = 0, where a, b & c are real numbers and the coefficients of x and y, i.e a and b respectively, are not equal to zero.

For example: 

• a₁x + b₁y + c₁ = 0
• a₂x + b₂y + c₂ = 0

Read More: Arithmetic Progressions Revision Notes

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Formulas for Quadratic Equations

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For any quadratic equation, algebraic formula is given as: 

Let an algebraic equation, ax² + bx + c = 0,

So, the formula for this quadratic equation will be: 

(α,ß) = [-b ± √(b² – 4ac)]/2ac

• The quadratic equation will have two distinct real roots if b²- 4ac >0
• The quadratic equation will have two imaginary roots if b² – 4ac < 0
• The quadratic equation has two real equivalent real roots if b² – 4ac = 0

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Progression Formulas

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Progression formulas are given below

• The nth term of an arithmetic sequence (an) = a + (n-1)d
• The sum of n terms in an arithmetic sequence (sn) = n/2 [2a + (n-1)d]
• The nth term of a geometric sequence (an) = a.rⁿ-1
• The sum of n terms of a geometric sequence (sn) = a(1-rⁿ)/(1-r)
• The sum of infinite terms of a geometric sequence (s) = a/(1-r)

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Things to Remember

• A single equation without the equal sign is known as an algebraic expression whereas a set of equations with the equal sign is known as the algebraic equation.
• An equation that has a mixture of variables and is only true for some value of x is known as an algebraic equation.
• A polynomial of 1 degree is known as linear, 2 degrees is known as quadratic, and degree 3 is known as a cubic polynomial.
• A quadratic equation has 2-3 zeros which form the x coordinators of the points where the graph of y=p(x) intersects the x-axis.