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The sum of two irrational numbers multiplied by the larger one is 70 and their difference is multiplied by the smaller one is12;the two numbers are?
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The sum of two irrational numbers multiplied by the larger one is 70 a...
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The sum of two irrational numbers multiplied by the larger one is 70 a...
Solution:
Let the two irrational numbers be x and y, where x>y.
Given,
x + y = 70/x ...(1)
x - y = 12/y ...(2)
Multiplying equation (1) by x and equation (2) by y, we get
x² + xy = 70 ...(3)
xy - y² = 12 ...(4)
Adding equations (3) and (4), we get
x² + xy + xy - y² = 70 + 12
x² + 2xy - y² = 82
(x+y)² - 4xy = 82 [Using (x+y)² = x² + 2xy + y²]
(70/x)² - 4xy = 82
4900/x² - 4xy = 82
4900 - 4xy² = 82x² [Multiplying throughout by x²]
4xy² + 82x² - 4900 = 0
Using the quadratic formula, we get
xy = [ -82 ± √(82² + 4(4)(4900)) ] / 8
xy = [ -82 ± 410 ] / 8
Since x and y are both irrational, we can discard the negative value of xy. Therefore,
xy = (410-82)/8 = 39
Now, using equation (1),
x + y = 70/x
y = 70/x - x
Substituting y in terms of x in equation (4), we get
x(70/x - x) - y² = 12
70 - x³ - y²x = 12x
x³ + y²x - 12x - 70 = 0
Substituting xy = 39, we get
x³ + 39x - 70x - 70 = 0
x³ - 31x - 70 = 0
Using trial and error method, we get x = 7.28 (approx.) and y = 62.72 (approx.)
Therefore, the two irrational numbers are 7.28 and 62.72.
Let the two irrational numbers be x and y, where x>y.
Given,
x + y = 70/x ...(1)
x - y = 12/y ...(2)
Multiplying equation (1) by x and equation (2) by y, we get
x² + xy = 70 ...(3)
xy - y² = 12 ...(4)
Adding equations (3) and (4), we get
x² + xy + xy - y² = 70 + 12
x² + 2xy - y² = 82
(x+y)² - 4xy = 82 [Using (x+y)² = x² + 2xy + y²]
(70/x)² - 4xy = 82
4900/x² - 4xy = 82
4900 - 4xy² = 82x² [Multiplying throughout by x²]
4xy² + 82x² - 4900 = 0
Using the quadratic formula, we get
xy = [ -82 ± √(82² + 4(4)(4900)) ] / 8
xy = [ -82 ± 410 ] / 8
Since x and y are both irrational, we can discard the negative value of xy. Therefore,
xy = (410-82)/8 = 39
Now, using equation (1),
x + y = 70/x
y = 70/x - x
Substituting y in terms of x in equation (4), we get
x(70/x - x) - y² = 12
70 - x³ - y²x = 12x
x³ + y²x - 12x - 70 = 0
Substituting xy = 39, we get
x³ + 39x - 70x - 70 = 0
x³ - 31x - 70 = 0
Using trial and error method, we get x = 7.28 (approx.) and y = 62.72 (approx.)
Therefore, the two irrational numbers are 7.28 and 62.72.
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The sum of two irrational numbers multiplied by the larger one is 70 and their difference is multiplied by the smaller one is12;the two numbers are?
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The sum of two irrational numbers multiplied by the larger one is 70 and their difference is multiplied by the smaller one is12;the two numbers are? for CA Foundation 2024 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about The sum of two irrational numbers multiplied by the larger one is 70 and their difference is multiplied by the smaller one is12;the two numbers are? covers all topics & solutions for CA Foundation 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The sum of two irrational numbers multiplied by the larger one is 70 and their difference is multiplied by the smaller one is12;the two numbers are?.
The sum of two irrational numbers multiplied by the larger one is 70 and their difference is multiplied by the smaller one is12;the two numbers are? for CA Foundation 2024 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about The sum of two irrational numbers multiplied by the larger one is 70 and their difference is multiplied by the smaller one is12;the two numbers are? covers all topics & solutions for CA Foundation 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The sum of two irrational numbers multiplied by the larger one is 70 and their difference is multiplied by the smaller one is12;the two numbers are?.
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