Algebra Full Explaining part 2

 

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• Quadratic Equations: A quadratic equation can be written in the standard form as ax² + bx + c = 0, where a, b, c are constants and x is the variable. The values of x that satisfy the equation are called solutions of the equation, and a quadratic equation has at most two solutions.
• Cubic Equations: The algebraic equations having variables with power 3 are referred to as cubic equations. A generalized form of a cubic equation is ax³ + bx² + cx + d = 0. A cubic equation has numerous applications in calculus and three-dimensional geometry (3D Geometry).

Sequence and Series

A set of numbers having a relationship across the numbers is called a sequence. A sequence is a set of numbers having a common mathematical relationship between the number, and a series is the sum of the terms of a sequence. In mathematics, we have two broad number sequences and series in the form of arithmetic progression and geometric progression. Some of these series are finite and some series are infinite. The two series are also called arithmetic progression and geometric progression and can be represented as follows.

• Arithmetic Progression: An Arithmetic progression (AP) is a special type of progression in which the difference between two consecutive terms is always a constant. The terms of an arithmetic progression series is a, a+d, a + 2d, a + 3d, a + 4d, a + 5d, .....
• Geometric Progression: Any progression in which the ratio of adjacent terms is fixed is a Geometric Progression. The general form of representation of a geometric sequence is a, ar, ar², ar³, ar⁴, ar⁵, .....

Exponents

Exponent is a mathematical operation, written as aⁿ. Here the expression aⁿ involves two numbers, the base 'a' and the exponent or power 'n'. Exponents are used to simplify algebraic expressions. In this section, we are going to learn in detail about exponents including squares, cubes, square root, and cube root. The names are based on the powers of these exponents. The exponents can be represented in the form aⁿ = a × a × a × ... n times.

Logarithms

The logarithm is the inverse function to exponents in algebra. Logarithms are a convenient way to simplify large algebraic expressions. The exponential form represented as a^x = n can be transformed into logarithmic form as log${}_a$n = x. John Napier discovered the concept of Logarithms in 1614. Logarithms have now become an integral part of modern mathematics.

Sets

A set is a well-defined collection of distinct objects and is used to represent algebraic variables. The purpose of using sets is to represent the collection of relevant objects in a group. Example: Set A = {2, 4, 6, 8}..........(A set of even numbers), Set B = {a, e, i, o, u}......(A set of vowels).

Algebraic Formulas

An algebraic identity is an equation that is always true regardless of the values assigned to the variables. Identity means that the left-hand side of the equation is identical to the right-hand side, for all values of the variables. These formulae involve squares and cubes of algebraic expressions and help in solving the algebraic expressions in a few quick steps. The frequently used algebraic formulas are listed below.

• (a + b)² = a² + 2ab + b²
• (a - b)² = a² - 2ab + b²
• (a + b)(a - b) = a² - b²
• (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
• (a + b)³ = a³ + 3a²b + 3ab² + b³
• (a - b)³ = a³ - 3a²b + 3ab² - b³

Let us see the application of these formulas in algebra using the following example,

Example: Using the (a + b)² formula in algebra, find the value of (101)².

Solution:

Given: (101)² = (100 + 1)²
Using algebra formula (a + b)² = a² + 2ab + b², we have,
(100 + 1)² = (100)² + 2(1)(100) + (1)²
(101)² = 10201

For more formulas check the page of algebraic formulas, containing the formulas for expansion of algebraic expressions, exponents, and logarithmic formulas.

Algebraic Operations

The basic operations covered in algebra are addition, subtraction, multiplication, and division.

• Addition: For the addition operation in algebra, two or more expressions are separated by a plus (+) sign between them.
• Subtraction: For the subtraction operation in algebra, two or more expressions are separated by a minus (-) sign between them.
• Multiplication: For the multiplication operation in algebra, two or more expressions are separated by a multiplication (×) sign between them.
• Division: For the division operation in algebra, two or more expressions are separated by a "/" sign between them.

Basic Rules and Properties of Algebra

The basic rules or properties of algebra for variables, algebraic expressions, or real numbers a, b and c are as given below,

• Commutative Property of Addition: a + b = b + a
• Commutative Property of Multiplication: a × b = b × a
• Associative Property of Addition: a + (b + c) = (a + b) + c
• Associative Property of Multiplication: a × (b × c) = (a × b) × c
• Distributive Property: a × (b + c) = (a × b) + (a × c), or, a × (b - c) = (a × b) - (a × c)
• Reciprocal: Reciprocal of a = 1/a
• Additive Identity Property: a + 0 = 0 + a = a
• Multiplicative Identity Property: a × 1 = 1 × a = a
• Additive Inverse: a + (-a) = 0

• Examples on Algebra

  1. Example 1: Expand (2x + 3y)² using the algebraic identities.

     Solution:

     Here we shall use an identity in algebra, (a + b)² = a² + 2ab + b²

     (2x + 3y)² = (2x)² + 2(2x)(3y) + (3y)²

     = 4x² + 12xy + 9y²

     Therefore, the answer is (2x + 3y)² = 4x² + 12xy + 9y²

  2. Example 2: The present age of a person is double the age of his son. Ten years ago, his age was four times the age of his son. Use the concept of algebra and find the present age of the son.

     Solution:

     Let us consider the present age of the son as 'x' years. It is given that the age of the person is double the age of his son, so the age of the person is '2x' years. Now considering the situation 10 years ago, the age of the son was (x - 10) years and the age of the person was (2x - 10) years. The question says that 10 years ago the age of the person was 4 times the age of his son. Therefore, this can be expressed as,

     2x - 10 = 4(x - 10)

     2x - 10 = 4x - 40

     2x - 4x = -40 + 10

     -2x = -30

     2x = 30

     x = 30/2

     x = 15

     Therefore, the present age of the son is 15 years.

  3. Example 3: Five less than a number equals to two. What is the number?

     Solution:

     Using the concept of Algebra, we will assume the number to be a variable. Let the number be x. As per the question, we can write x - 5 = 2. On solving this, we get x = 7. Therefore, the required number is 7.