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CLOSURE PROPERTY (DIVISION OF WHOLE NUMBERS) AT ALGEBRA DEN Before understanding this topic you must know what are whole numbers ? Explanation :-System of whole numbers is not closed under division, this means that the division of any two whole numbers is not always a whole number. This is known as Closure Property for Division of Whole Numbers. Read the following terms and you can further understand thisproperty
CONSTRUCTION OF 135 DEGREE ANGLE WITH THE HELP OF COMPASS Construction of 135 Degree Angle with the help of Compass : math, algebra & geometry tutorials for school and home education A - B WHOLE SQUARE AT ALGEBRA DEN Example 1: Solve (3a - 2b) 2 Solution: This proceeds as: Given polynomial (3a - 2b) 2 represents identity second i.e. (a - b) 2 Where a = 3a and b = 2b Now apply values of a and b on the identity i.e. (a - b) 2 = a 2 + b 2 - 2ab and we get: (3a - 2b) 2 = (3a) 2 + (2b) 2 - 2(3a)(2b) Expand the exponential forms and we get: A + B WHOLE SQUARE AT ALGEBRA DEN Following are few applications of identity first: Example 1: Solve (4p + 5q) 2 Solution: This proceeds as: Given polynomial (4p + 5q) 2 represents identity first i.e. (a + b) 2 Where a = 4p and b = 5q DIFFERENCE & SIMILARITY BETWEEN RECTANGLE & PARALLELOGRAM Hence, you can observe that there 6 similarities between a Rectangle and a Parallelogram.The highlighted part is the difference between a Rectangle and a Parallelogram. LENGTHS OF TWO SIDES OF A TRIANGLE ARE 7 CM AND 9 CM. CAN Lengths of two sides of a triangle are 7 cm and 9 cm. Can you guess between which two numbers will the length of third side fall ? As we know that Sum of two sides of a triangle is greater than the thirdside.
FACTORS, HCF, LCM, FRACTIONS, INTEGERS Factors, HCF, LCM, Fractions, Integers, Ratio Proportion, Constants, Multiples, Numbers, Arithmetic Expression : math, algebra & geometry tutorials for school and (A + B + C) WHOLE SQUARE AT ALGEBRA DEN Example 1: Solve (4p + 5q + 3r) 2 Solution: This proceeds as: Given polynomial (4p + 5q + 3r) 2 represents identity first i.e. (a + b+ c) 2 Where a = 4p, b = 5q and c = 3r Now apply values of a, b and c on the identity i.e. (a + b +c) 2 = a 2 + b 2 + c 2 + 2ab + 2bc + 2ca and we get: (4p + 5q + 3r) 2 = (4p) 2 + (5q) 2 + (3r) 2 + 2(4p)(5q) + 2(5q)(3r) + 2(3r)(4p) Expand the exponential forms CONSTRUCTION OF 45 DEGREE ANGLE WITH THE HELP OF COMPASS To construct 45 degree angle, first we draw 90 degree angle and its done in the following steps: 1). Use ruler and draw a Line segment OB of any convenient length. (as shown below) 2). Now use compass and open it to any convenient radius. CONSTRUCTION OF 90 DEGREE ANGLE WITH THE HELP OF COMPASS Follow the following step to construct 90 Degree Angle 1). Use ruler and draw a Line segment OB of any convenient length. (as shown below) 2). Now use compass and open it to any convenient radius. And with O as center , draw an arc which cuts line segment OB at X. (as shownbelow)
CLOSURE PROPERTY (DIVISION OF WHOLE NUMBERS) AT ALGEBRA DEN Before understanding this topic you must know what are whole numbers ? Explanation :-System of whole numbers is not closed under division, this means that the division of any two whole numbers is not always a whole number. This is known as Closure Property for Division of Whole Numbers. Read the following terms and you can further understand thisproperty
CONSTRUCTION OF 135 DEGREE ANGLE WITH THE HELP OF COMPASS Construction of 135 Degree Angle with the help of Compass : math, algebra & geometry tutorials for school and home education A - B WHOLE SQUARE AT ALGEBRA DEN Example 1: Solve (3a - 2b) 2 Solution: This proceeds as: Given polynomial (3a - 2b) 2 represents identity second i.e. (a - b) 2 Where a = 3a and b = 2b Now apply values of a and b on the identity i.e. (a - b) 2 = a 2 + b 2 - 2ab and we get: (3a - 2b) 2 = (3a) 2 + (2b) 2 - 2(3a)(2b) Expand the exponential forms and we get: A + B WHOLE SQUARE AT ALGEBRA DEN Following are few applications of identity first: Example 1: Solve (4p + 5q) 2 Solution: This proceeds as: Given polynomial (4p + 5q) 2 represents identity first i.e. (a + b) 2 Where a = 4p and b = 5q DIFFERENCE & SIMILARITY BETWEEN RECTANGLE & PARALLELOGRAM Hence, you can observe that there 6 similarities between a Rectangle and a Parallelogram.The highlighted part is the difference between a Rectangle and a Parallelogram. LENGTHS OF TWO SIDES OF A TRIANGLE ARE 7 CM AND 9 CM. CAN Lengths of two sides of a triangle are 7 cm and 9 cm. Can you guess between which two numbers will the length of third side fall ? As we know that Sum of two sides of a triangle is greater than the thirdside.
FACTORS, HCF, LCM, FRACTIONS, INTEGERS Factors, HCF, LCM, Fractions, Integers, Ratio Proportion, Constants, Multiples, Numbers, Arithmetic Expression : math, algebra & geometry tutorials for school and CONSTRUCTION OF SQUARE WITH COMPASS AT ALGEBRA DEN Construction of Square with Compass : math, algebra & geometry tutorials for school and home education POLYNOMIALS CAN HAVE COEFFICIENTS, TERMS, VARIABLES AND Polynomials can have Coefficients, Terms, Variables and involves Powers of the Variable : math, algebra & geometry tutorials for schooland home education
FRACTION IN DIAGRAMMATIC FORM AT ALGEBRA DEN Fraction cans be represented in the form of diagram also. e.g. In the following diagram; there is a rectangle which is divided into twoequal parts.
CONSTRUCTION OF 135 DEGREE ANGLE WITH THE HELP OF COMPASS Construction of 135 Degree Angle with the help of Compass : math, algebra & geometry tutorials for school and home education DIFFERENCE & SIMILARITY BETWEEN RECTANGLE & PARALLELOGRAM Hence, you can observe that there 6 similarities between a Rectangle and a Parallelogram.The highlighted part is the difference between a Rectangle and a Parallelogram. DIFFERENCE & SIMILARITY BETWEEN SQUARE & RECTANGLE AT Following points helps us to understand similarities and differences between a square and a rectangle. The highlighted part is the difference between square and rectangle. LENGTHS OF TWO SIDES OF A TRIANGLE ARE 7 CM AND 9 CM. CAN Lengths of two sides of a triangle are 7 cm and 9 cm. Can you guess between which two numbers will the length of third side fall ? As we know that Sum of two sides of a triangle is greater than the thirdside.
DIFFERENCE & SIMILARITY BETWEEN RHOMBUS & SQUARE AT Hence you can observe the similarities between Rhombus and Square. The highlighted part is the difference between a Rhombus and Square RAMON HAS A HORSE AND A CARABAO. THE HORSE IS 2/3 AS OLD Ramon has a horse and a Carabao. the horse is 2/3 as old as the carabao. The difference between their ages is 4 years. How old is the carabao? : math, algebra & FACTORS, HCF, LCM, FRACTIONS, INTEGERS Factors, HCF, LCM, Fractions, Integers, Ratio Proportion, Constants, Multiples, Numbers, Arithmetic Expression : math, algebra & geometry tutorials for school and (A + B + C) WHOLE SQUARE AT ALGEBRA DEN Example 1: Solve (4p + 5q + 3r) 2 Solution: This proceeds as: Given polynomial (4p + 5q + 3r) 2 represents identity first i.e. (a + b+ c) 2 Where a = 4p, b = 5q and c = 3r Now apply values of a, b and c on the identity i.e. (a + b +c) 2 = a 2 + b 2 + c 2 + 2ab + 2bc + 2ca and we get: (4p + 5q + 3r) 2 = (4p) 2 + (5q) 2 + (3r) 2 + 2(4p)(5q) + 2(5q)(3r) + 2(3r)(4p) Expand the exponential forms A - B WHOLE SQUARE AT ALGEBRA DEN Multiply as we do multiplication of two binomials and we get: = a (a - b) - b (a - b) = a 2 - ab - ab + b 2. Add like terms and we get: = a 2 - 2ab + b 2. Rearrange the terms and we get: = a 2 + b 2 - 2ab. Hence, in this way we obtain the identity i.e. (a - b) 2 = a 2 + b 2 - 2ab. Following are few applications this identity. (A - B) CUBE AT ALGEBRA DEN Multiply trinomial with binomial as shown below: = a 2 (a - b) + b 2 (a - b) - 2ab (a - b) = a 3 - a 2 b + ab 2 - b 3 - 2a 2 b + 2ab 2. Rearrange the terms and we get: = a 3 - b 3 - a2b - 2a2b + ab2 + 2ab2. Add like terms, highlighted in green & red and we get: = a 3 - b 3 - 3a2b + 3ab2. Or we can further solve it: A + B WHOLE SQUARE AT ALGEBRA DEN Following are few applications of identity first: Example 1: Solve (4p + 5q) 2 Solution: This proceeds as: Given polynomial (4p + 5q) 2 represents identity first i.e. (a + b) 2 Where a = 4p and b = 5q IN A TOWN 35% ARE MALES, 25% ARE FEMALES. FIND % OF Sol:- As per given in the question -. Males in a town = 35%. Females in a town = 25%. Total population of a town is sum of male, female and children. Or, we can also write as. Total population = male% + females% + children %. Now, we know that the value of total population is 100% , because all parts when join together form a whole. CONSTRUCTION OF 90 DEGREE ANGLE WITH THE HELP OF COMPASS Follow the following step to construct 90 Degree Angle. 1). Use ruler and draw a Line segment OB of any convenient length. (as shown below) 2). Now use compass and open it to any convenient radius. And with O as center , draw an arc which cuts line segment OB at X. (as shown below) 3). Again use compass and opened to the same radius (as of step2).
LENGTHS OF TWO SIDES OF A TRIANGLE ARE 7 CM AND 9 CM. CAN Lengths of two sides of a triangle are 7 cm and 9 cm. Can you guess between which two numbers will the length of third side fall ? As we know that Sum of two sides of a triangle is greater than the thirdside.
IN A CLASS OF 40 STUDENTS, IF GIRLS ARE 40%. FIND TOTAL Hence, number of boys in class is 24. Method 2: Find percentage of boys and then find number of boys: Percentage of girls = 40% of total students. Percentage of Boys = 100% - percentage of girls. Put value of percentage of girls and we get: Percentage of Boys = 100% - 40%. Solve subtraction expression and we get: Percentage of Boys = 60%. AT A POINT 10M AWAY FROM THE FOOT OF A BUILDING, THE ANGLE At a point 10m away from the foot of a building, the angle of elevation of the top of building is 60° find the height of building : Let AB is the building. FACTORS, HCF, LCM, FRACTIONS, INTEGERS Factors, HCF, LCM, Fractions, Integers, Ratio Proportion, Constants, Multiples, Numbers, Arithmetic Expression : math, algebra & geometry tutorials for school and (A CUBE + B CUBE + C CUBE) Example: Solve 8a 3 + 27b 3 + 125c 3 - 30abc Solution: This proceeds as: Given polynomial (8a 3 + 27b 3 + 125c 3 - 30abc) can be written as: (2a) 3 + (3b) 3 + (5c) 3 - (2a)(3b)(5c) And this represents identity: a 3 + b 3 + c 3 - 3abc = (a + b + c)(a 2 + b 2 + c 2 - ab - bc - ca) Where a = 2a, b = 3b and c = 5c Now apply values of a, b and c on the L.H.S of identity i.e. a 3 + b 3 + c 3 - 3abc A + B WHOLE SQUARE AT ALGEBRA DEN Following are few applications of identity first: Example 1: Solve (4p + 5q) 2 Solution: This proceeds as: Given polynomial (4p + 5q) 2 represents identity first i.e. (a + b) 2 Where a = 4p and b = 5q (A + B + C) WHOLE SQUARE AT ALGEBRA DEN Example 1: Solve (4p + 5q + 3r) 2 Solution: This proceeds as: Given polynomial (4p + 5q + 3r) 2 represents identity first i.e. (a + b+ c) 2 Where a = 4p, b = 5q and c = 3r Now apply values of a, b and c on the identity i.e. (a + b +c) 2 = a 2 + b 2 + c 2 + 2ab + 2bc + 2ca and we get: (4p + 5q + 3r) 2 = (4p) 2 + (5q) 2 + (3r) 2 + 2(4p)(5q) + 2(5q)(3r) + 2(3r)(4p) Expand the exponential forms A - B WHOLE SQUARE AT ALGEBRA DEN Multiply as we do multiplication of two binomials and we get: = a (a - b) - b (a - b) = a 2 - ab - ab + b 2. Add like terms and we get: = a 2 - 2ab + b 2. Rearrange the terms and we get: = a 2 + b 2 - 2ab. Hence, in this way we obtain the identity i.e. (a - b) 2 = a 2 + b 2 - 2ab. Following are few applications this identity. IN A TOWN 35% ARE MALES, 25% ARE FEMALES. FIND % OF Sol:- As per given in the question -. Males in a town = 35%. Females in a town = 25%. Total population of a town is sum of male, female and children. Or, we can also write as. Total population = male% + females% + children %. Now, we know that the value of total population is 100% , because all parts when join together form a whole. CONSTRUCTION OF 90 DEGREE ANGLE WITH THE HELP OF COMPASS Follow the following step to construct 90 Degree Angle. 1). Use ruler and draw a Line segment OB of any convenient length. (as shown below) 2). Now use compass and open it to any convenient radius. And with O as center , draw an arc which cuts line segment OB at X. (as shown below) 3). Again use compass and opened to the same radius (as of step2).
LENGTHS OF TWO SIDES OF A TRIANGLE ARE 7 CM AND 9 CM. CAN Lengths of two sides of a triangle are 7 cm and 9 cm. Can you guess between which two numbers will the length of third side fall ? As we know that Sum of two sides of a triangle is greater than the thirdside.
IN A CLASS OF 40 STUDENTS, IF GIRLS ARE 40%. FIND TOTAL Hence, number of boys in class is 24. Method 2: Find percentage of boys and then find number of boys: Percentage of girls = 40% of total students. Percentage of Boys = 100% - percentage of girls. Put value of percentage of girls and we get: Percentage of Boys = 100% - 40%. Solve subtraction expression and we get: Percentage of Boys = 60%. AT A POINT 10M AWAY FROM THE FOOT OF A BUILDING, THE ANGLE At a point 10m away from the foot of a building, the angle of elevation of the top of building is 60° find the height of building : Let AB is the building. CONSTRUCTION OF 90 DEGREE ANGLE WITH THE HELP OF COMPASS Follow the following step to construct 90 Degree Angle. 1). Use ruler and draw a Line segment OB of any convenient length. (as shown below) 2). Now use compass and open it to any convenient radius. And with O as center , draw an arc which cuts line segment OB at X. (as shown below) 3). Again use compass and opened to the same radius (as of step2).
NATURAL NUMBERS AT ALGEBRA DEN Answer = From the given series pick and separate Zero and negative numbers and we get. Natural Numbers = 23, 55, 890, 34, 323. Example 2 : From the given series of numbers, find natural numbers. Given series = 4, (0.43), 45, 900, -9, 0, -47. Answer = From the given series pick and separate Zero, negative numbers and decimal numbers and we get. LITERAL NUMBERS AT ALGEBRA DEN Definition In Algebra, we many times uses English Alphabets like a, x, y, z to represent a variable. So, we call all such English Alphabets as Literal Numbers. In simple words, an English Alphabet that is used to represent a variable is called a Literal Number. From the following algebraic numbers pick out Literal Numbers. MULTIPLICATION OF MONOMIAL & TRINOMIAL AT ALGEBRA DEN While multiplying a monomial and a trinomial you will find following situations: Multiply Monomial and Trinomial. Example 1: Multiply 5a and (a + b + c) Example 2: Multiply 3p and (p 2 + p - 2) Multiply Trinomial and Monomial. Example 3: Multiply (4q 2 - 2q + 8) and 2q. Example 4: Multiply (a + b - c) and z. Multiply Monomial andTrinomial.
FINDING ROOTS OF QUADRATIC EQUATION OR SOLUTION OF The formula to find the roots of a quadratic equation is known as the Quadratic Formula: Note: if b 2 - 4ac > 0 then only we can find the roots of quadratic equation with this formula. Let's understand how we get this formula There are two methods in this: CONSTRUCTION OF 135 DEGREE ANGLE WITH THE HELP OF COMPASS To construct 135 degree angle we first construct 90 degree angle and its steps of constructions are as follows: 1). Use ruler and draw a Line segment OB of any convenient length. (as shown below) 2). Now use compass and open it to any convenient radius. And with O as center , draw an arc which cuts line segment OB at X . (as shown below) 3). ADDITION OF BINOMIALS AT ALGEBRA DEN Addition of Binomials having like terms is done in the following steps: Step 1: Arrange the binomials in like terms. Step 2: Add like terms. Example 1: Add 12ab + 10 and 10ab + 5. Solution: Given two binomials: First Binomial = 12ab + 10. Second Binomial = 10ab + 5. Now addition of given binomials is LENGTHS OF TWO SIDES OF A TRIANGLE ARE 7 CM AND 9 CM. CAN Lengths of two sides of a triangle are 7 cm and 9 cm. Can you guess between which two numbers will the length of third side fall ? As we know that Sum of two sides of a triangle is greater than the thirdside.
IN A CLASS OF 40 STUDENTS, IF GIRLS ARE 40%. FIND TOTAL Hence, number of boys in class is 24. Method 2: Find percentage of boys and then find number of boys: Percentage of girls = 40% of total students. Percentage of Boys = 100% - percentage of girls. Put value of percentage of girls and we get: Percentage of Boys = 100% - 40%. Solve subtraction expression and we get: Percentage of Boys = 60%. DIFFERENCE & SIMILARITY BETWEEN RECTANGLE & PARALLELOGRAM Diagonals of rectangle bisect each other. Diagonals of parallelogram also bisect each other. Opposite sides of rectangle are of equal length. Opposite sides of parallelogram are also of equal length. All angles of a Rectangle are also of 90 Degree each i.e. right angled at each vertex. Angles of parallelogram can be of different degree. FACTORS, HCF, LCM, FRACTIONS, INTEGERS Factors, HCF, LCM, Fractions, Integers, Ratio Proportion, Constants, Multiples, Numbers, Arithmetic Expression : math, algebra & geometry tutorials for school and (A + B + C) WHOLE SQUARE AT ALGEBRA DEN Example 1: Solve (4p + 5q + 3r) 2 Solution: This proceeds as: Given polynomial (4p + 5q + 3r) 2 represents identity first i.e. (a + b+ c) 2 Where a = 4p, b = 5q and c = 3r Now apply values of a, b and c on the identity i.e. (a + b +c) 2 = a 2 + b 2 + c 2 + 2ab + 2bc + 2ca and we get: (4p + 5q + 3r) 2 = (4p) 2 + (5q) 2 + (3r) 2 + 2(4p)(5q) + 2(5q)(3r) + 2(3r)(4p) Expand the exponential forms A - B WHOLE SQUARE AT ALGEBRA DEN Multiply as we do multiplication of two binomials and we get: = a (a - b) - b (a - b) = a 2 - ab - ab + b 2. Add like terms and we get: = a 2 - 2ab + b 2. Rearrange the terms and we get: = a 2 + b 2 - 2ab. Hence, in this way we obtain the identity i.e. (a - b) 2 = a 2 + b 2 - 2ab. Following are few applications this identity. (A - B) CUBE AT ALGEBRA DEN Multiply trinomial with binomial as shown below: = a 2 (a - b) + b 2 (a - b) - 2ab (a - b) = a 3 - a 2 b + ab 2 - b 3 - 2a 2 b + 2ab 2. Rearrange the terms and we get: = a 3 - b 3 - a2b - 2a2b + ab2 + 2ab2. Add like terms, highlighted in green & red and we get: = a 3 - b 3 - 3a2b + 3ab2. Or we can further solve it: A + B WHOLE SQUARE AT ALGEBRA DEN Following are few applications of identity first: Example 1: Solve (4p + 5q) 2 Solution: This proceeds as: Given polynomial (4p + 5q) 2 represents identity first i.e. (a + b) 2 Where a = 4p and b = 5q IN A TOWN 35% ARE MALES, 25% ARE FEMALES. FIND % OF Sol:- As per given in the question -. Males in a town = 35%. Females in a town = 25%. Total population of a town is sum of male, female and children. Or, we can also write as. Total population = male% + females% + children %. Now, we know that the value of total population is 100% , because all parts when join together form a whole. CONSTRUCTION OF 90 DEGREE ANGLE WITH THE HELP OF COMPASS Follow the following step to construct 90 Degree Angle. 1). Use ruler and draw a Line segment OB of any convenient length. (as shown below) 2). Now use compass and open it to any convenient radius. And with O as center , draw an arc which cuts line segment OB at X. (as shown below) 3). Again use compass and opened to the same radius (as of step2).
LENGTHS OF TWO SIDES OF A TRIANGLE ARE 7 CM AND 9 CM. CAN Lengths of two sides of a triangle are 7 cm and 9 cm. Can you guess between which two numbers will the length of third side fall ? As we know that Sum of two sides of a triangle is greater than the thirdside.
IN A CLASS OF 40 STUDENTS, IF GIRLS ARE 40%. FIND TOTAL Hence, number of boys in class is 24. Method 2: Find percentage of boys and then find number of boys: Percentage of girls = 40% of total students. Percentage of Boys = 100% - percentage of girls. Put value of percentage of girls and we get: Percentage of Boys = 100% - 40%. Solve subtraction expression and we get: Percentage of Boys = 60%. AT A POINT 10M AWAY FROM THE FOOT OF A BUILDING, THE ANGLE At a point 10m away from the foot of a building, the angle of elevation of the top of building is 60° find the height of building : Let AB is the building. FACTORS, HCF, LCM, FRACTIONS, INTEGERS Factors, HCF, LCM, Fractions, Integers, Ratio Proportion, Constants, Multiples, Numbers, Arithmetic Expression : math, algebra & geometry tutorials for school and (A + B + C) WHOLE SQUARE AT ALGEBRA DEN Example 1: Solve (4p + 5q + 3r) 2 Solution: This proceeds as: Given polynomial (4p + 5q + 3r) 2 represents identity first i.e. (a + b+ c) 2 Where a = 4p, b = 5q and c = 3r Now apply values of a, b and c on the identity i.e. (a + b +c) 2 = a 2 + b 2 + c 2 + 2ab + 2bc + 2ca and we get: (4p + 5q + 3r) 2 = (4p) 2 + (5q) 2 + (3r) 2 + 2(4p)(5q) + 2(5q)(3r) + 2(3r)(4p) Expand the exponential forms A - B WHOLE SQUARE AT ALGEBRA DEN Multiply as we do multiplication of two binomials and we get: = a (a - b) - b (a - b) = a 2 - ab - ab + b 2. Add like terms and we get: = a 2 - 2ab + b 2. Rearrange the terms and we get: = a 2 + b 2 - 2ab. Hence, in this way we obtain the identity i.e. (a - b) 2 = a 2 + b 2 - 2ab. Following are few applications this identity. (A - B) CUBE AT ALGEBRA DEN Multiply trinomial with binomial as shown below: = a 2 (a - b) + b 2 (a - b) - 2ab (a - b) = a 3 - a 2 b + ab 2 - b 3 - 2a 2 b + 2ab 2. Rearrange the terms and we get: = a 3 - b 3 - a2b - 2a2b + ab2 + 2ab2. Add like terms, highlighted in green & red and we get: = a 3 - b 3 - 3a2b + 3ab2. Or we can further solve it: A + B WHOLE SQUARE AT ALGEBRA DEN Following are few applications of identity first: Example 1: Solve (4p + 5q) 2 Solution: This proceeds as: Given polynomial (4p + 5q) 2 represents identity first i.e. (a + b) 2 Where a = 4p and b = 5q IN A TOWN 35% ARE MALES, 25% ARE FEMALES. FIND % OF Sol:- As per given in the question -. Males in a town = 35%. Females in a town = 25%. Total population of a town is sum of male, female and children. Or, we can also write as. Total population = male% + females% + children %. Now, we know that the value of total population is 100% , because all parts when join together form a whole. CONSTRUCTION OF 90 DEGREE ANGLE WITH THE HELP OF COMPASS Follow the following step to construct 90 Degree Angle. 1). Use ruler and draw a Line segment OB of any convenient length. (as shown below) 2). Now use compass and open it to any convenient radius. And with O as center , draw an arc which cuts line segment OB at X. (as shown below) 3). Again use compass and opened to the same radius (as of step2).
LENGTHS OF TWO SIDES OF A TRIANGLE ARE 7 CM AND 9 CM. CAN Lengths of two sides of a triangle are 7 cm and 9 cm. Can you guess between which two numbers will the length of third side fall ? As we know that Sum of two sides of a triangle is greater than the thirdside.
IN A CLASS OF 40 STUDENTS, IF GIRLS ARE 40%. FIND TOTAL Hence, number of boys in class is 24. Method 2: Find percentage of boys and then find number of boys: Percentage of girls = 40% of total students. Percentage of Boys = 100% - percentage of girls. Put value of percentage of girls and we get: Percentage of Boys = 100% - 40%. Solve subtraction expression and we get: Percentage of Boys = 60%. AT A POINT 10M AWAY FROM THE FOOT OF A BUILDING, THE ANGLE At a point 10m away from the foot of a building, the angle of elevation of the top of building is 60° find the height of building : Let AB is the building. CONSTRUCTION OF 90 DEGREE ANGLE WITH THE HELP OF COMPASS Follow the following step to construct 90 Degree Angle. 1). Use ruler and draw a Line segment OB of any convenient length. (as shown below) 2). Now use compass and open it to any convenient radius. And with O as center , draw an arc which cuts line segment OB at X. (as shown below) 3). Again use compass and opened to the same radius (as of step2).
NATURAL NUMBERS AT ALGEBRA DEN Answer = From the given series pick and separate Zero and negative numbers and we get. Natural Numbers = 23, 55, 890, 34, 323. Example 2 : From the given series of numbers, find natural numbers. Given series = 4, (0.43), 45, 900, -9, 0, -47. Answer = From the given series pick and separate Zero, negative numbers and decimal numbers and we get. LITERAL NUMBERS AT ALGEBRA DEN Definition In Algebra, we many times uses English Alphabets like a, x, y, z to represent a variable. So, we call all such English Alphabets as Literal Numbers. In simple words, an English Alphabet that is used to represent a variable is called a Literal Number. From the following algebraic numbers pick out Literal Numbers. MULTIPLICATION OF MONOMIAL & TRINOMIAL AT ALGEBRA DEN While multiplying a monomial and a trinomial you will find following situations: Multiply Monomial and Trinomial. Example 1: Multiply 5a and (a + b + c) Example 2: Multiply 3p and (p 2 + p - 2) Multiply Trinomial and Monomial. Example 3: Multiply (4q 2 - 2q + 8) and 2q. Example 4: Multiply (a + b - c) and z. Multiply Monomial andTrinomial.
FINDING ROOTS OF QUADRATIC EQUATION OR SOLUTION OF The formula to find the roots of a quadratic equation is known as the Quadratic Formula: Note: if b 2 - 4ac > 0 then only we can find the roots of quadratic equation with this formula. Let's understand how we get this formula There are two methods in this: CONSTRUCTION OF 135 DEGREE ANGLE WITH THE HELP OF COMPASS To construct 135 degree angle we first construct 90 degree angle and its steps of constructions are as follows: 1). Use ruler and draw a Line segment OB of any convenient length. (as shown below) 2). Now use compass and open it to any convenient radius. And with O as center , draw an arc which cuts line segment OB at X . (as shown below) 3). ADDITION OF BINOMIALS AT ALGEBRA DEN Addition of Binomials having like terms is done in the following steps: Step 1: Arrange the binomials in like terms. Step 2: Add like terms. Example 1: Add 12ab + 10 and 10ab + 5. Solution: Given two binomials: First Binomial = 12ab + 10. Second Binomial = 10ab + 5. Now addition of given binomials is LENGTHS OF TWO SIDES OF A TRIANGLE ARE 7 CM AND 9 CM. CAN Lengths of two sides of a triangle are 7 cm and 9 cm. Can you guess between which two numbers will the length of third side fall ? As we know that Sum of two sides of a triangle is greater than the thirdside.
IN A CLASS OF 40 STUDENTS, IF GIRLS ARE 40%. FIND TOTAL Hence, number of boys in class is 24. Method 2: Find percentage of boys and then find number of boys: Percentage of girls = 40% of total students. Percentage of Boys = 100% - percentage of girls. Put value of percentage of girls and we get: Percentage of Boys = 100% - 40%. Solve subtraction expression and we get: Percentage of Boys = 60%. DIFFERENCE & SIMILARITY BETWEEN RECTANGLE & PARALLELOGRAM Diagonals of rectangle bisect each other. Diagonals of parallelogram also bisect each other. Opposite sides of rectangle are of equal length. Opposite sides of parallelogram are also of equal length. All angles of a Rectangle are also of 90 Degree each i.e. right angled at each vertex. Angles of parallelogram can be of different degree. (A + B + C) WHOLE SQUARE AT ALGEBRA DEN Example 1: Solve (4p + 5q + 3r) 2 Solution: This proceeds as: Given polynomial (4p + 5q + 3r) 2 represents identity first i.e. (a + b+ c) 2 Where a = 4p, b = 5q and c = 3r Now apply values of a, b and c on the identity i.e. (a + b +c) 2 = a 2 + b 2 + c 2 + 2ab + 2bc + 2ca and we get: (4p + 5q + 3r) 2 = (4p) 2 + (5q) 2 + (3r) 2 + 2(4p)(5q) + 2(5q)(3r) + 2(3r)(4p) Expand the exponential forms DIFFERENCE & SIMILARITY BETWEEN SQUARE & RECTANGLE AT All angles of a Square are of 90 Degree each i.e. right angled at each vertex. All angles of a Rectangle are also of 90 Degree each i.e. right angled at each vertex. Opposite sides of square are parallel to each other. Opposite sides of rectangle are also parallel to each other. Diagonals of square are equal. Diagonals of rectangle are alsoequal.
MULTIPLICATION OF MONOMIAL & TRINOMIAL AT ALGEBRA DENBINOMIAL TRINOMIAL MONOMIAL CALCULATORCONSTANT MONOMIALMONOMIAL VS BINOMIAL VS TRINOMIALPOLYNOMIAL BINOMIAL TRINOMIAL While multiplying a monomial and a trinomial you will find following situations: Multiply Monomial and Trinomial. Example 1: Multiply 5a and (a + b + c) Example 2: Multiply 3p and (p 2 + p - 2) Multiply Trinomial and Monomial. Example 3: Multiply (4q 2 - 2q + 8) and 2q. Example 4: Multiply (a + b - c) and z. Multiply Monomial andTrinomial.
DIFFERENCE & SIMILARITY BETWEEN RECTANGLE & PARALLELOGRAM Diagonals of rectangle bisect each other. Diagonals of parallelogram also bisect each other. Opposite sides of rectangle are of equal length. Opposite sides of parallelogram are also of equal length. All angles of a Rectangle are also of 90 Degree each i.e. right angled at each vertex. Angles of parallelogram can be of different degree. CLOSURE PROPERTY (DIVISION OF WHOLE NUMBERS) AT ALGEBRA DEN Before understanding this topic you must know what are whole numbers ? Explanation :-System of whole numbers is not closed under division, this means that the division of any two whole numbers is not always a whole number. This is known as Closure Property for Division of Whole Numbers. Read the following terms and you can further understand thisproperty
ARRANGE FOLLOWING DECIMALS IN ASCENDING ORDER: 0.42, 0.27ARRANGING DECIMALS IN ASCENDING ORDERORDER NUMBERS IN ASCENDING ORDERPUT NUMBERS IN ASCENDING ORDERASCENDING ORDER IN MATHASCENDING ORDER NUMBERHOW TO ORDER DECIMAL NUMBERS So 0.42 is written next to decimals 0.27 in the ascending order and we get series: Ascending Order Series = 0.15, 0.27, 0.42. Step 4: Lastly, we are left with only one decimal, whose digit at tenths place is the largest among digits at tenths place of all the given decimals and it would be written at the last place of the order: Since, decimal INTERIOR OPPOSITE ANGLES OF TRIANGLE AT ALGEBRA DENALTERNATE INTERIOR ANGLES FORMULAOPPOSITE INTERIOR ANGLES THEOREMALTERNATE ANGLES MATHALTERNATE INTERIOR ANGLES EQUAL WHATOPPOSITE EXTERIOR ANGLESALTERNATE INTERIOR ANGLES PROBLEMS Exterior Angles of a Triangle. In the following diagram you can see exterior angles and interior angles. Let's find interior opposite angles in the following ways: ∠3 & ∠5 are Interior Opposite Angles to exterior ∠1, as shown in the following diagram: ∠5 & ∠4 are Interior Opposite Angles to exterior ∠DIFFERENCE & SIMILARITY BETWEEN RHOMBUS & SQUARE AT Diagonals of Rhombus are of unequal length. Diagonals of Square are of equal length. Only opposite angles of Rhombus are of equal measure. All angles of Square are of 90 degree each i.e. right angled at each Vertex. Hence you can observe the similarities between Rhombus and Square. The highlighted part is the difference between a Rhombus and IN A TOWN 35% ARE MALES, 25% ARE FEMALES. FIND % OF Sol:- As per given in the question -. Males in a town = 35%. Females in a town = 25%. Total population of a town is sum of male, female and children. Or, we can also write as. Total population = male% + females% + children %. Now, we know that the value of total population is 100% , because all parts when join together form a whole. 5 PACKS OF BISCUITS WEIGHT 500 GRAMS. WHAT IS THE WEIGHT 5 packs of biscuits weight 500 grams. What is the weight of 12 biscuits packs. This proceeds: Weight of 5 biscuits packs = 500grams Weight of 1 biscuit pack = 500 ÷ 5 = 100grams (A + B + C) WHOLE SQUARE AT ALGEBRA DEN Example 1: Solve (4p + 5q + 3r) 2 Solution: This proceeds as: Given polynomial (4p + 5q + 3r) 2 represents identity first i.e. (a + b+ c) 2 Where a = 4p, b = 5q and c = 3r Now apply values of a, b and c on the identity i.e. (a + b +c) 2 = a 2 + b 2 + c 2 + 2ab + 2bc + 2ca and we get: (4p + 5q + 3r) 2 = (4p) 2 + (5q) 2 + (3r) 2 + 2(4p)(5q) + 2(5q)(3r) + 2(3r)(4p) Expand the exponential forms DIFFERENCE & SIMILARITY BETWEEN SQUARE & RECTANGLE AT All angles of a Square are of 90 Degree each i.e. right angled at each vertex. All angles of a Rectangle are also of 90 Degree each i.e. right angled at each vertex. Opposite sides of square are parallel to each other. Opposite sides of rectangle are also parallel to each other. Diagonals of square are equal. Diagonals of rectangle are alsoequal.
MULTIPLICATION OF MONOMIAL & TRINOMIAL AT ALGEBRA DENBINOMIAL TRINOMIAL MONOMIAL CALCULATORCONSTANT MONOMIALMONOMIAL VS BINOMIAL VS TRINOMIALPOLYNOMIAL BINOMIAL TRINOMIAL While multiplying a monomial and a trinomial you will find following situations: Multiply Monomial and Trinomial. Example 1: Multiply 5a and (a + b + c) Example 2: Multiply 3p and (p 2 + p - 2) Multiply Trinomial and Monomial. Example 3: Multiply (4q 2 - 2q + 8) and 2q. Example 4: Multiply (a + b - c) and z. Multiply Monomial andTrinomial.
DIFFERENCE & SIMILARITY BETWEEN RECTANGLE & PARALLELOGRAM Diagonals of rectangle bisect each other. Diagonals of parallelogram also bisect each other. Opposite sides of rectangle are of equal length. Opposite sides of parallelogram are also of equal length. All angles of a Rectangle are also of 90 Degree each i.e. right angled at each vertex. Angles of parallelogram can be of different degree. CLOSURE PROPERTY (DIVISION OF WHOLE NUMBERS) AT ALGEBRA DEN Before understanding this topic you must know what are whole numbers ? Explanation :-System of whole numbers is not closed under division, this means that the division of any two whole numbers is not always a whole number. This is known as Closure Property for Division of Whole Numbers. Read the following terms and you can further understand thisproperty
ARRANGE FOLLOWING DECIMALS IN ASCENDING ORDER: 0.42, 0.27ARRANGING DECIMALS IN ASCENDING ORDERORDER NUMBERS IN ASCENDING ORDERPUT NUMBERS IN ASCENDING ORDERASCENDING ORDER IN MATHASCENDING ORDER NUMBERHOW TO ORDER DECIMAL NUMBERS So 0.42 is written next to decimals 0.27 in the ascending order and we get series: Ascending Order Series = 0.15, 0.27, 0.42. Step 4: Lastly, we are left with only one decimal, whose digit at tenths place is the largest among digits at tenths place of all the given decimals and it would be written at the last place of the order: Since, decimal INTERIOR OPPOSITE ANGLES OF TRIANGLE AT ALGEBRA DENALTERNATE INTERIOR ANGLES FORMULAOPPOSITE INTERIOR ANGLES THEOREMALTERNATE ANGLES MATHALTERNATE INTERIOR ANGLES EQUAL WHATOPPOSITE EXTERIOR ANGLESALTERNATE INTERIOR ANGLES PROBLEMS Exterior Angles of a Triangle. In the following diagram you can see exterior angles and interior angles. Let's find interior opposite angles in the following ways: ∠3 & ∠5 are Interior Opposite Angles to exterior ∠1, as shown in the following diagram: ∠5 & ∠4 are Interior Opposite Angles to exterior ∠DIFFERENCE & SIMILARITY BETWEEN RHOMBUS & SQUARE AT Diagonals of Rhombus are of unequal length. Diagonals of Square are of equal length. Only opposite angles of Rhombus are of equal measure. All angles of Square are of 90 degree each i.e. right angled at each Vertex. Hence you can observe the similarities between Rhombus and Square. The highlighted part is the difference between a Rhombus and IN A TOWN 35% ARE MALES, 25% ARE FEMALES. FIND % OF Sol:- As per given in the question -. Males in a town = 35%. Females in a town = 25%. Total population of a town is sum of male, female and children. Or, we can also write as. Total population = male% + females% + children %. Now, we know that the value of total population is 100% , because all parts when join together form a whole. 5 PACKS OF BISCUITS WEIGHT 500 GRAMS. WHAT IS THE WEIGHT 5 packs of biscuits weight 500 grams. What is the weight of 12 biscuits packs. This proceeds: Weight of 5 biscuits packs = 500grams Weight of 1 biscuit pack = 500 ÷ 5 = 100grams FACTORS, HCF, LCM, FRACTIONS, INTEGERS Factors, HCF, LCM, Fractions, Integers, Ratio Proportion, Constants, Multiples, Numbers, Arithmetic Expression : math, algebra & geometry tutorials for school and A - B WHOLE SQUARE AT ALGEBRA DEN Multiply as we do multiplication of two binomials and we get: = a (a - b) - b (a - b) = a 2 - ab - ab + b 2. Add like terms and we get: = a 2 - 2ab + b 2. Rearrange the terms and we get: = a 2 + b 2 - 2ab. Hence, in this way we obtain the identity i.e. (a - b) 2 = a 2 + b 2 - 2ab. Following are few applications this identity. A + B WHOLE SQUARE AT ALGEBRA DEN Following are few applications of identity first: Example 1: Solve (4p + 5q) 2 Solution: This proceeds as: Given polynomial (4p + 5q) 2 represents identity first i.e. (a + b) 2 Where a = 4p and b = 5q NATURAL NUMBERS AT ALGEBRA DEN Example 3 : From the given series of numbers, find natural numbers. Given series = 43, 4/5, 0, (0.987), 20, -67, -7, 2 Answer = From the given series pick and separate Zero, negative numbers, fractional numbers and decimal numbers and we get Natural Numbers = 43, 20, 2. DIFFERENCE & SIMILARITY BETWEEN SQUARE & RECTANGLE AT All angles of a Square are of 90 Degree each i.e. right angled at each vertex. All angles of a Rectangle are also of 90 Degree each i.e. right angled at each vertex. Opposite sides of square are parallel to each other. Opposite sides of rectangle are also parallel to each other. Diagonals of square are equal. Diagonals of rectangle are alsoequal.
CONSTRUCTION OF 90 DEGREE ANGLE WITH THE HELP OF COMPASS Follow the following step to construct 90 Degree Angle. 1). Use ruler and draw a Line segment OB of any convenient length. (as shown below) 2). Now use compass and open it to any convenient radius. And with O as center , draw an arc which cuts line segment OB at X. (as shown below) 3). Again use compass and opened to the same radius (as of step2).
DIFFERENCE & SIMILARITY BETWEEN SQUARE & PARALLELOGRAM AT A square is a quadrilateral. A parallelogram is also a quadrilateral. Opposite sides of square are parallel to each other. Opposite sides of parallelogram are also parallel to each other. Opposite angles of square are equal. Opposite angles of parallelogram are also equal. Adjacent angles of square are supplementary. DIFFERENCE & SIMILARITY BETWEEN RHOMBUS & SQUARE AT Diagonals of Rhombus are of unequal length. Diagonals of Square are of equal length. Only opposite angles of Rhombus are of equal measure. All angles of Square are of 90 degree each i.e. right angled at each Vertex. Hence you can observe the similarities between Rhombus and Square. The highlighted part is the difference between a Rhombus and IN A TOWN 35% ARE MALES, 25% ARE FEMALES. FIND % OF Sol:- As per given in the question -. Males in a town = 35%. Females in a town = 25%. Total population of a town is sum of male, female and children. Or, we can also write as. Total population = male% + females% + children %. Now, we know that the value of total population is 100% , because all parts when join together form a whole. 5 PACKS OF BISCUITS WEIGHT 500 GRAMS. WHAT IS THE WEIGHT 5 packs of biscuits weight 500 grams. What is the weight of 12 biscuits packs. This proceeds: Weight of 5 biscuits packs = 500grams Weight of 1 biscuit pack = 500 ÷ 5 = 100gramsAlgebra Den
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