Algebra Formulas- Detailed Definition & Formulas

algebra formulas
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Introduction to Algebra

 

In this algebra formulas article, we will provide you with a detailed list of Algebraic Expressions in Maths, their definition, and examples. This article will be helpful for all the students who want to get better in Mathematics. You can refer to these algebra formulas provided here while solving the questions.

 

Algebra is a branch of mathematics that substitutes letters for numbers. An algebraic equation depicts a scale, what is done on one side of the scale with a number is also done to either side of the scale. The numbers are constants. Algebra also includes real numbers, complex numbers, matrices, vectors and much more. X, Y, A, B are the most commonly used letters that represent algebraic problems and equations.

 

Mathematics is a vast field. It is impossible for one person to know everything there is to know in mathematics, even after a lifetime of study. And while it can be cumbersome, mathematics is also one of the most important fields of study. Right from how much to tip the waiter to when the universe began, all answers can be found due to the application of maths.

 

As we approach the higher classes, we see our introduction to algebra. In algebra, we substitute numbers with letters or alphabets to arrive at a solution. We use these letters like (x, a, b etc.) to represent unknown quantities in an equation. Then we solve the equation or algebra formula to arrive at a definite answer.

 

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Algebra itself is divided into two major fields. The more basic functions that we learn in school are known as elementary algebra. Then the more advanced algebra formula, which is more abstract in nature fall under modern algebra, sometimes even known as abstract algebra.

 

Download Handwritten written Algebra Formulas

 

Math Symbols

 

If you would like to create your own math expressions, here are some symbols that the calculator understands:

+ (Addition)
– (Subtraction)
* (Multiplication)
/ (Division)
^ (Exponent: “raised to the power”)
sqrt (Square Root) (Example: sqrt(9))

 

What is an Equation?

An equation says that two things are equal. It will have an equals sign “=” like this:

x+2=6

That equations says: what is on the left (x + 2) is equal to what is on the right (6)

So an equation is like a statement “this equals that”

(Note: this equation has the solution x=4)

 

Algebra includes both numbers and letters. Numbers are fixed, i.e. their value is known. Letters or alphabets are used to represent the unknown quantities in the algebra formula. Now, a combination of numbers, letters, factorials, matrices etc. is used to form an equation or formula. This is essentially the methodology for algebra.

 

What is a Formula?

 

A formula is a fact or rule that uses mathematical symbols.

It will usually have:

an equals sign (=)

two or more variables (x, y, etc) that stand in for values we don’t know yet

It shows us how things are related to each other.

 

As students study for their exams, there are certain very important algebra formulas and equations that they must learn. These formulas are the cornerstone of basic or elementary algebra. Only learning the formulas is not sufficient. The students must also understand the concept behind the formula and learn to apply them correctly.

 

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Algebra is represented as the study of unknown quantities. Its concepts become important for students studying from Class 6 to the higher classes. This article on Algebra Formulas and Expression covers the following topics.

 

Algebra Formulas: Important Algebraic Identities

 

Algebraic identities comprise various equality equations consisting of different variables.

 

a) Linear Equations in One Variable: A linear equation in one variable has the maximum of one variable present in the order 1. It is depicted in the form of ax + b = 0, where x is represented as the variable.

 

b) Linear Equations in Two Variables: A linear equation in two variables consists of the utmost two variables present in order 2. The equation is depicted in the form: ax2 + bx + c = 0. The two variables are quite important because your coursebook has a lot of questions based on it. So, you need to stay focused on important algebra formulas to find the solution.

 

Here, we will provide a list of all the important algebra formulas. The comprehensive list will allow the students to have a quick look before exams or refer to whenever they wish. Remember, only rote learning is not enough. You must also know how to effectively apply these formulas to a problem.

 

Some basic identities to note are:

The combination of literal numbers obeys every basic rule of addition, subtraction, multiplication and division.

x × y = xy; such as 5 × a = 5a = a × 5.

a × a × a × … 9 more times = a12

If a number is x8, then x is the base and 8 is the exponent.

A constant is a symbol with a fixed numerical value.

 

Algebra Formulas

 

  1. Binomial Theorem

    1. a2 – b2 = (a – b)(a + b)
    2. (a + b)2 = a2 + 2ab + b2
    3. a2 + b2 = (a + b)2 – 2ab
    4. (a – b)2 = a2 – 2ab + b2
    5. (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
    6. (a – b – c)2 = a2 + b2 + c2 – 2ab + 2bc – 2ca
    7. (a + b)3 = a3 + 3a2b + 3ab2 + b3 ; (a + b)3 = a3 + b3 + 3ab(a + b)
    8. (a – b)3 = a3 – 3a2b + 3ab2 – b3 = a3 – b3 – 3ab(a – b)
    9. a3 – b3 = (a – b)(a2 + ab + b2)
    10. a3 + b3 = (a + b)(a2 – ab + b2)
    11. (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4
    12. (a – b)4 = a4 – 4a3b + 6a2b2 – 4ab3 + b4
    13. a4 – b4 = (a – b)(a + b)(a2 + b2)
    14. a5 – b5 = (a – b)(a4 + a3b + a2b2 + ab3 + b4)
    15. If n is a natural number an – bn = (a – b)(an-1 + an-2b+…+ bn-2a + bn-1)
    16. If n is even (n = 2k), an + bn = (a + b)(an-1 – an-2b +…+ bn-2a – bn-1)
    17. If n is odd (n = 2k + 1), an + bn = (a + b)(an-1 – an-2b +an-3b2…- bn-2a + bn-1)
    18. (a + b + c + …)2 = a2 + b2 + c2 + … + 2(ab + ac + bc + ….)
    19. Laws of Exponents (am)(an) = am+n ; (ab)m = ambm ; (am)n = amn

 

Fractional Exponents a0 = 1

algebra formulas

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And so on. These formulas are easy to remember because of their symmetry and these are used very frequently in Algebra. We will cover a number of examples that involve these formulas.

 

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2. Difference of Two Squares Formula

 

x2−y2=(x+y)(x−y)x2-y2=(x+y)(x-y)

 

This formula indicates that we can determine the difference of two squares simply my taking a product of the sum and the difference of the variables involved.

 

3. Sum / Difference of Two Cubes

 

x3+y3=(x+y)(x2−xy+y2)x3+y3=(x+y)(x2-xy+y2)

x3−y3=(x−y)(x2+xy+y2)x3-y3=(x-y)(x2+xy+y2)

 

Roots of Quadratic Equation

 

For a quadratic equation ax2 + bx + c = 0 where a ≠ 0, the roots will be given by the equation as x=−b±b2−4ac√2a

Δ = b2 − 4ac is called the discriminant

For real and distinct roots, Δ > 0

For real and coincident roots, Δ = 0

For non-real roots, Δ < 0

If α and β are the two roots of the equation ax2 + bx + c = 0 then, α + β = (-b / a) and α × β = (c / a).

If the roots of a quadratic equation are α and β, the equation will be (x − α)(x − β) = 0

Factorials

n! = (1).(2).(3)…..(n − 1).n

n! = n(n − 1)! = n(n − 1)(n − 2)! = ….

0! = 1

 

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Solved Examples:- 

 

Q: Find the value of 17² – 4²

Ans: Now these are simple numbers, so we can calculate the answer. But the correct method is to apply the formula,

a² – b² = (a-b)(a+b)

17² – 4² = (17-4)(17+4) = 13 × 21 = 273

 

Q: Find out the value of 52 – 32

Ans: Using the formula a2 – b2 = (a – b)(a + b)
where a = 5 and b = 3
(a – b)(a + b)
= (5 – 3)(5 + 3)
= 2 × 8
= 16

 

Q: 43 × 42 = ?

Ans: Using the exponential formula (am)(an) = am+n
where a = 4
43 × 42
= 43+2
= 45
= 1024

 

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Frequently Asked Questions (FAQs): 

 

Q: Why Algebra is considered important in Mathematics?

Ans: Algebra is an important concept in applied mathematics. It is undeniably the best component that can help you understand the theory of partial differential equations. These are quite important in physical systems such as movement and forces as well as heat transfers, and more. Therefore, to stay clear of these physical aspects, you need to be proficient with the basics of algebra formulas and equations.

 

Q: What are the different components of the Algebra formulas and expressions?

Ans: Algebra formulas and expressions can be divided into the following components:
1. Algebraic Identities
2. Laws of Exponent
3. Quadratic Equations
4. Other Important Expressions

 

 

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