A
Learning Task 5: Solve this in your notebook.
1. Suppose that you friend factors x2y + 36xy like this:
24x3y + 36xy
4xy(6x + 9)
= (4xy)(3)(2x + 3)
= 12xy(2x + 3)
Is this correct? Would you suggest any changes?
2. If the sum of the square of the number and 4 time the number is 21,
what is the number?
3. The length of the rectangle is 3 less than twice the width. If the area is 9
square ft, the length and width of the rectangle.
4. Given the figure below. Find the area of the shaded part of the rectangle if the
area of the big rectangle is 6 times the area of the unshaded rectangle
2x
3
x 3
Answers
Mathematics (from Greek: μάθημα, máthēma, 'knowledge, study, learning') includes the study of such topics as quantity (number theory),[1] structure (algebra),[2] space (geometry),[1] and change (mathematical analysis).[3][4][5] It has no generally accepted definition.[6][7]
Mathematicians seek and use patterns[8][9] to formulate new conjectures; they resolve the truth or falsity of such by mathematical proof. When mathematical structures are good models of real phenomena, mathematical reasoning can be used to provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.
Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements.[10] Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.[11]
Mathematics is essential in many fields, including natural science, engineering, medicine, finance, and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics (mathematics for its own sake) without having any application in mind, but practical applications for what began as pure mathematics are often discovered later.[12][13]
Answer:
1. It is incorrect. Correct solution given below.
2. The number is either 3 or -7
3. Length of the given rectangle = 3 ft
Breadth of the given rectangle = 3 ft
4. The area of shaded part is 10x²
Step-by-step explanation:
1. Given solution of factorizing x²y + 36xy is given by:
24x³y + 36xy
= 4xy(6x + 9)
= (4xy)(3)(2x + 3)
= 12xy(2x + 3)
This is wrong
Because we have to factorize x²y + 36xy and not 24x³y + 36xy
The correct steps will be as under:
x²y + 36xy
= xy(x + 36)
2. Let the number be 'n'
According to the conditions of the question,
=> n² + 4n = 21
=> n² + 4n - 21 = 0
=> n² + 7n - 3n - 21 = 0
=> n(n + 7) -3(n + 7) = 0
=> (n + 7) (n - 3) = 0
=> n + 7 = 0 or n - 3 = 0
=> n = - 7 or n = 3
Therefore, the number is either 3 or -7
3. Let the length of the rectangle be 'l' and breadth be 'b'
According to the conditions of the question,
=> l = 2*b - 3
=> l = 2b - 3
Given that area = 9 ft² = Length * Breadth
=> 9 = (2b - 3)b
=> 9 = 2b² - 3b
=> 2b² - 3b - 9 = 0
=> 2b² - 6b + 3b - 9 = 0
=> 2b(b - 3) +3(b - 3) = 0
=> (2b + 3) (b - 3) = 0
=> 2b + 3 = 0 or b - 3 = 0
=> b = -3/2 or b = 3
Since breadth cannot be negative,
b = 3 ft = Breadth
As l = 2b - 3
= 2*3 - 3
= 6 - 3
= 3
l = 3 ft = Length
Therefore, length of the given rectangle = 3 ft and breadth of the given rectangle = 3 ft
4. Area of shaded part = Area of outer rectangle - Area of inner rectangle
According to the question,
Area of big rectangle = 6 * Area of inner rectangle
Area of shaded part = 6*Area of inner rectangle - Area of inner rectangle
= 5 * Area of inner rectangle
= 5(2x*x)
= 5(2x²)
= 10x²
Therefore, the area of shaded part is 10x²