## Algebra Formulas in Maths

Algebra is a branch of mathematics that deals with the manipulation and use of variables, symbols, and equations to solve mathematical problems. Algebra Formulas have a vital role to play in solving algebraic equations, and they are used in various fields such as science, engineering, economics, and finance. **Algebra is taught to children from school to college (if opt for the engineering field) and competitive examinations like SSC, Banking, RRB, etc**. It is said that Algebra is the stepping stone to your success and a firm grasp of Algebraic expressions and formulae helps to excel in your career.

In this article, we will cover some of the most important algebra formulas and expressions that are commonly used in mathematics and other fields.

## Algebraic Expressions Formula

**Algebraic Expression** is defined as the mathematical expression made up of variables and constants, in addition to algebraic operations like multiplication, subtraction, addition, etc. There are **mainly three kinds of Algebraic Expression** as explained given below.

**Monomial Expression-**

This Algebraic Expression has only a single term like 2x, 6y, etc.

**Binomial Expression-**

This Algebraic Expression has two terms like 6xy+5, xy+ y², etc.

**Polynomial Expression-**

This Algebraic Expression has more than two terms with the non-negative integral exponents of a variable like 6x²+4x+7, 3y³+5y+15, etc.

## Basic Algebra Formulas

An algebraic expression or Equation is made when the vectors, numbers, letters, and matrices are combinedly and used in Algebra Formulas. In a general view of the Algebraic Expression, the value of the number used in the equation is known but the value of the letter used is unknown. Hence Algebra Formulas are applied to find out the values of unknown quantities. **The Basic Algebra Formulas are mentioned below in the table**.

### Basic Algebra Formulas

- (a+b)² = a² + 2ab + b²
- (a-b)² = a² – 2ab + b²
- a² – b² = (a-b)(a+b)
- a² + b² = (a-b)² +2ab
- (a+b+c)² = a²+b²+c²+2ab+2ac+2bc
- (a-b-c)² = a²+b²+c²-2ab-2ac+2bc
- (a+b)³ = a³+ 3a²b + 3ab² + b³
- (a-b)³ = a³- b³ + 3ab² - 3a²b
- a³-b³ = (a² + ab + b²)(a - b)
- a³+b³ = (a² – ab + b²)(a + b)
- (a+b) (a-b) = a2 – b2
- (x + a)(x + b) = x2 + (a + b)x + ab
- (x + a)(x – b) = x2 + (a – b)x – ab
- (x – a)(x + b) = x2 + (b – a)x – ab
- (x – a)(x – b) = x2 – (a + b)x + ab
- (a + b)3 = a3 + b3 + 3ab(a + b)
- (a – b)3 = a3 – b3 – 3ab(a – b)
- (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2xz
- (x + y – z)2 = x2 + y2 + z2 + 2xy – 2yz – 2xz
- (x – y + z)2 = x2 + y2 + z2 – 2xy – 2yz + 2xz
- (x – y – z)2 = x2 + y2 + z2 – 2xy + 2yz – 2xz
- x3 + y3 + z3 – 3xyz = (x + y + z)(x2 + y2 + z2 – xy – yz -xz)
- x2 + y2 =½ [(x + y)2 + (x – y)2]
- (x + a) (x + b) (x + c) = x3 + (a + b +c)x2 + (ab + bc + ca)x + abc
- x3 + y3= (x + y) (x2 – xy + y2)
- x3 – y3 = (x – y) (x2 + xy + y2)
- x2 + y2 + z2 -xy – yz – zx = ½ [(x-y)2 + (y-z)2 + (z-x)2]

## Algebra Formula Chart

**Algebra** is a mathematical study that deals with geometry, number theory, and analysis. It has a vast reach to cover from solving elementary equations to the study of abstractions. Algebra Formulas have always been an important topic for almost all competitive examinations. **The complete list of Algebra Formulas and Expressions is tabulated below.**

Algebra Formulas Chart | |

1. | a⁴ – b⁴ = (a² + b²) (a² – b²) |

2. | a⁵ – b⁵ = (a – b)(a⁴+ a³b + a²b² + ab³ + b⁴ ) |

3. | a⁵ + b⁵ = (a + b)(a⁴ – a³b + a²b²– ab³ + b⁴ ) |

4. | a³ + b³+ c³– 3abc = (a + b + c)(a² + b² + c² – ab – bc – ca) |

5. | (a + b + c+...)² = a²+b²+c²+...+2(ab + bc+....) |

6. | If n is a natural number, a^n − b^n = (a−b)(a^(n−1) + a^(n−2) b+...+b^(n−2) a + b^(n−1)) |

7. | If n is even (n=2k), a^n + b^n = (a+b)(a^(n−1) − a^(n−2) b+...+b^(n−2) a − b^(n−1)) |

8. | If n is odd (n=2k+1), a^n + b^n = (a+b)(a^(n−1) − a^(n−2) b +...−b^(n−2) a + b^(n−1)) |

9. | (x+y+z)²=x²+y²+z²+2xy+2yz+2xz |

10. | (x+y−z)²=x²+y²+z²+2xy−2yz−2xz |

11. | (x−y+z)²=x²+y²+z²−2xy−2yz+2xz |

12. | (x−y−z)²=x²+y²+z²−2xy+2yz−2xz |

13. | x³+y³+z³−3xyz=(x+y+z)(x²+y²+z²−xy−yz−xz) |

14. | (x+a)(x+b)(x+c)=x³+(a+b+c)x²+(ab+bc+ca)x+abc |

15. | x²+y²+z²−xy−yz−zx=1/2[(x−y)²+(y−z)²+(z−x)²] |

- Quadratic Formula: The quadratic formula is used to solve quadratic equations of the form ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable.

The formula is as follows: x = (-b ± √(b^2 - 4ac)) / 2a

This formula gives the solutions for x that satisfy the equation. If the expression inside the square root sign is negative, the equation has no real solutions.

- Distance Formula: The distance formula is used to find the distance between two points in a coordinate plane. If the coordinates of the two points are (x1, y1) and (x2, y2), then the distance formula is as follows:

d = √((x2 - x1)^2 + (y2 - y1)^2)

This formula can also be used in three-dimensional space by adding a third coordinate (z) and using the Pythagorean theorem.

- Slope Formula: The slope formula is used to find the slope of a line passing through two points (x1, y1) and (x2, y2). The formula is as follows:

m = (y2 - y1) / (x2 - x1)

The slope of a line gives the rate of change of y with respect to x, and it is represented by the letter m.

- Exponential Formula: The exponential formula is used to represent exponential growth or decay. If a quantity A grows exponentially over time t at a rate of r, then the formula is as follows:

A = A0 e^(rt)

where A0 is the initial quantity and e is the base of the natural logarithm.

If a quantity decays exponentially over time t at a rate of r, then the formula is as follows:

A = A0 e^(-rt)

where A0 is the initial quantity.

- Logarithmic Formula: The logarithmic formula is used to represent the inverse of exponential growth or decay. If y = b^x, then the logarithmic formula is as follows:

logb(y) = x

This formula gives the exponent x that satisfies the equation y = b^x. The base of the logarithm can be any positive number, but commonly used bases are 10 and e.

- Pythagorean Theorem: The Pythagorean theorem is used to find the length of the sides of a right triangle. If a and b are the lengths of the two legs of a right triangle and c is the length of the hypotenuse, then the formula is as follows:

c^2 = a^2 + b^2

This formula can be used to find any of the three lengths if the other two are known.

- Factorial Formula: The factorial formula is used to find the factorial of a number. The factorial of a positive integer n is the product of all positive integers from 1 to n. The formula is as follows:

n! = 1 * 2 * 3 * ... * n

For example, 5! = 1 * 2 * 3 * 4 * 5 = 120.

These are some of the most important algebra formulas that are commonly used in mathematics and other fields. By understanding and applying these formulas, you can solve a wide range of Algebraic problems.

**Laws of Exponents-**

Some **basic laws of exponents are used** while calculating complex exponential expressions. Students do not need to expand the exponential terms while solving the exponential expressions and they can easily compute the greater exponential values.

- (a^m)(a^n)=a^(m+n)
- (ab)^m=a^m b^m
- (a^m)^n=a^(mn)

**Fractional Exponents-**

- a^0=1
- a^m / a^n = a^ (m−n)
- a^m = 1/a^(−m)
- a^(−m) = 1/a^m

## Algebra Formulas Related Questions

Some of the important questions related to Algebra are mentioned below. These questions will help you a lot for a better understanding of the Algebra concepts.

**Question 1: Find out the Algebraic Expression 8y+6 when y=3 **

**Solution:** Putting the value of y in the given expression we get,

8 (3)+6 = 30

**Question 2: A total 47 number of boys are there in a class. The number of boys is three more than four times the number of girls in that class. How many girls are present in the class?**

**Solution:** Let the number of girls in the class be y

According to the question, the expression to be

No. of Boys in the class = 3 + 4y = 47

4y = 44

So the number of girls in the class is y = (44/4) = 11

**Question 3: Find the value of y in (y-1)² = [4√(y-4)]²**

**Solution:** y²-2y+1 = 1Triangle6(y-4)

y²-2y+1 = 16y-64

y²-18y+65 = 0

(y-13) (y-5) = 0

Hence the value of y = 13 and y = 5