Algebra is a branch of mathematics. It includes numbers and letters/alphabets.
Some basic algebra formulas are given below:-
a²–b² = (a–b)(a+b)
(a+b)² = a²+2ab+b²
a²+b² = (a+b)²–2ab
(a–b)² = a²–2ab+b²
(a+b+c)² = a²+b²+c²+2ab+2bc+2ca
(a–b–c)² = a²+b²+c²–2ab+2bc–Luca
(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)
a³+b³+c³ – 3abc = (a+b+c) (a²+b²+c²-ab-bc-ca)
(a+b)⁴ = a⁴+4a³b+6a²b²+4ab³+b⁴
(a-b)⁴ = a⁴-4a³b+6a²b²-4ab³+b⁴
a⁴-b⁴ = (a-b) (a+b) (a²+b²)
If (a+b+c) = 0 , then (a³+b³+c³) = 3abc
Some examples are given below:-
Question 1:- (a + 5)²
Ans:- (a + 5)²
= a² + 2 × a × 5 + 5²
= a² + 10a + 25
Question 2:- (5a + 4b)²
Ans:- (5a + 4b)²
= (5a)² + 2 × (5a) × (4b) + (4b)²
= 25a² + 40ab + 16b²
Question 3:- (3p³ + 4q²)²
Ans:- (3p³ + 4q²)²
= (3p³)² + 2 × (3p³) × (4q²) + (4q²)²
= 9p⁶ + 24p³q² + 16q⁴
Question 4:- (a – 7)²
Ans:- (a – 7)²
= a² – 2 × a × 7 × 7²
= a² – 14a + 49
Question 5:- (6p – 5)²
Ans:- (6p – 5)²
= (6p)² – 2 × 6p × 5 + (5)²
= 36p² – 60p + 25
Question 6:- (10p² – 3q)²
Ans:- (10p² -3q)²
= (10p²)² – 2 × 10p² × 3q + (3p)²
= 100p⁴ – 60p²q + 9q²
Question 7:- (a² – b²)²
Ans:- (a² – b²)²
= (a²)² – 2 × a² × b² + (b²)²
= a⁴ – 2a²b² + b⁴
Question 8:- (p + 12) (p – 12)
Ans:- (p +12) (p –12)
= p² – (12)²
= y² – 144
Question 9:- (2m + 3) (2m – 3)
Ans:- (2m + 3) (2m – 3)
= (2m)² – 3²
= 4m² – 9
Question 10:- (pq² – rs) (pq² + rs)
Ans:- (pq² – rs) (pq² + rs)
= (pq² + rs) (pq² – rs)
= (pq²)² – (rs)²
= p²q⁴ – r²s²
Question 11:- (a + 4)³
Ans:- (a + 4)³
= a³ + 3 × a² × 4 + 3 × a × 4² + 4³
= a³ + 12a² + 48a + 64
Question 12:- (5p + 2q²)³
Ans:- (5p)³ + 3 × (5p)² × 2q² + 3 × 5p × (2q²)² + (2q²)³
= 125p³ + 150p²q² + 60pq² + 8q⁶
Question 13:- (2p – 5)³
Ans:- (2p – 5)³
= (2p)³ – 3 × (2p)² × 5 + 3 × (2p) × 5² – 5³
= 8p³ – 60p² + 150p – 125
Question 14:- (3a² – 2b)³
Ans:- (3a² – 2b)³
= (3a²)³ – 3 × (3a²)² × 2b + 3 × 3a² × (2b)² – (2b)³
= 27a⁶ – 54a⁴b + 36a²b² – 8b³
Algebraic expression:-
Monomial
Binomial
Trinomial
Polynomial
Monomial:-
An algebraic expression with only one term is called Monomial. Examples:- 27x², 7xy, 6x³, 9xy etc.
Binomial:-
An algebraic expression with two unlike terms are called binomial signs. Examples:- 5x² + 7x, 32x + 4xy, 6x² – 9xy etc
Trinomial:-
An algebraic expression with three terms is called a trinomial. Examples:- 5x⁴ + 8xy + 2xy, 3x⁵y + 2x + 7xy etc
Polynomial:-
An algebraic expression with more than one term is called a polynomial. Examples:- 2x² + 4y³ + 2xy + xy² + y², 2x⁶ + 3xy + 5x⁴y + xy² etc
Algebra is a branch of mathematics which has a vast range of uses in different fields.
Here below given some uses of algebra :-
Solving Equations :-
Algebra is utilised to solve equations, such as linear equations, quadratic equations and exponential equations which are needed in many day to day activities, such as distances calculation, speeds measurement, and solving financial calculations.
Used in physics :-
Algebra is very much utilised in areas such as physics, where it is utilised to solve problems or calculations of motion, force, power, friction, electricity, and thermodynamics etc.
In software programs and data science :-
Algebraic formulas and concepts are widely utilised in software programming, algorithms and data science. It is used to solve problems and to process data.
In economical interpretation :-
Algebraic expressions are used while calculating interest, profit and loss, cost price analysis, sampling, data analytics, return on investments, and optimize resource allocation etc in economics.
Helpful in statistical analysis :-
Algebra is a method for working with data, finding averages, calculating regression models in the process of forecasting future trends, specially business organisations widely utilising these tools.
In space Science :-
Algebra is beneficial in calculating distances, velocities, and trajectories in space exploration, especially in the field of determination of the orbits of planets and spacecraft.
Algebra and its expression plays an important role in solving complex problems related to above mentioned topics, Its practical applicability is essential in various fields.
Apart from algebraic formulas, here some basic concepts of triangles are given below:-
Triangle:-
A triangle is a plane area of three angles with three sides.
Triangles can be divided into two bases.
On the basis of angle.
On the basis of sides.
On the basis of angle, there are three types of triangles.
Right triangle.
Acute triangle.
Obtuse triangle.
Right triangle:-
A triangle whose one of the angles is right angle i.e. 90°. Such a triangle is called a right-angled triangle.
Acute triangle:-
A triangle whose all the three angles are acute or less than 90° is called an acute triangle.
Obtuse triangle:-
A triangle whose one angle is greater than right angle i.e. 90°, is called an obtuse triangle.
On the basis of sides, there are three types of triangles.
Equilateral triangles.
Isosceles triangles.
Scalene triangles.
Equilateral Triangle:-
A triangle whose three sides are equal is called an equilateral triangle. In the case of an equilateral triangle, all three angles are equal i.e. 60°.
Isosceles triangle:-
A triangle whose two sides are equal is called an isosceles triangle.
Scalene triangles:-
A triangle whose all the three sides are different is called scalene triangles.
Angle :-
An angle is created by two rays with a common endpoint and the end point is called the vertex.
A protractor is used while measuring an angle.
The unit of measurement of an angle is degrees.
Types of Angles :-
There are seven types of angles.These are :-
Zero angle
Acute angle
Right angle
Obtuse angle
Straight angle
Reflex angle
Complete angle of full angle.
Zero Angle :-
A zero angle is an angle whose measure is exactly 0 degrees. This is created when the two rays which form the angle lie on the same line.
Acute Angle :-
An angle that measures less than 90 degrees but greater than 0 degrees is known as acute angle.
Right Angle :-
A right angle is an angle that measures exactly 90 degrees.
Obtuse Angle :-
An obtuse angle is an angle that measures more than 90 degrees but less than 180 degrees.
Straight Angle :-
A straight angle is an angle that measures exactly 180 degrees. It is formed when two rays create a straight line.
Reflex Angle :-
A reflex angle is an angle that measures greater than 180 degrees but less than 360 degrees.
Complete Angle or full angle :-
A complete angle or full angle is an angle that measures exactly 360 degrees.
These are basic algebra formulas, explanation of triangles and angles which will help students.
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