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Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation
- a)x2 + 18x + 16 = 0
- b)x2 – 18x – 16 = 0
- c)x2 + 18x – 16 = 0
- d)x2 – 18x + 16 = 0
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Let two numbers have arithmetic mean 9 and geometric mean 4. Then thes...
Let the two numbers be α and β. Given that
∴ Required equation is x2 – 18x + 16 = 0
∴ Required equation is x2 – 18x + 16 = 0
Most Upvoted Answer
Let two numbers have arithmetic mean 9 and geometric mean 4. Then thes...
Solution:
Given, arithmetic mean = 9 and geometric mean = 4.
Let the two numbers be a and b.
Arithmetic mean = (a + b)/2 = 9
Geometric mean = √ab = 4
Squaring both sides of the second equation, we get:
ab = 16
Multiplying both sides of the first equation by 2, we get:
a + b = 18
Squaring both sides of this equation, we get:
a2 + b2 + 2ab = 324
Substituting ab = 16, we get:
a2 + b2 + 32 = 324
a2 + b2 = 292
Multiplying both sides of the equation a + b = 18 by (a – b), we get:
a2 – b2 = 324 – 4ab
Substituting ab = 16, we get:
a2 – b2 = 260
Adding the equations a2 + b2 = 292 and a2 – b2 = 260, we get:
2a2 = 552
a2 = 276
a = √276 = 2√69
Substituting this value of a in the equation a + b = 18, we get:
2√69 + b = 18
b = 18 – 2√69
Therefore, the two numbers are:
a = 2√69 and b = 18 – 2√69
The quadratic equation with these roots is:
(x – 2√69)(x – (18 – 2√69)) = 0
Expanding this equation, we get:
x2 – 20x + 16 = 0
Hence, the correct answer is option D.
Given, arithmetic mean = 9 and geometric mean = 4.
Let the two numbers be a and b.
Arithmetic mean = (a + b)/2 = 9
Geometric mean = √ab = 4
Squaring both sides of the second equation, we get:
ab = 16
Multiplying both sides of the first equation by 2, we get:
a + b = 18
Squaring both sides of this equation, we get:
a2 + b2 + 2ab = 324
Substituting ab = 16, we get:
a2 + b2 + 32 = 324
a2 + b2 = 292
Multiplying both sides of the equation a + b = 18 by (a – b), we get:
a2 – b2 = 324 – 4ab
Substituting ab = 16, we get:
a2 – b2 = 260
Adding the equations a2 + b2 = 292 and a2 – b2 = 260, we get:
2a2 = 552
a2 = 276
a = √276 = 2√69
Substituting this value of a in the equation a + b = 18, we get:
2√69 + b = 18
b = 18 – 2√69
Therefore, the two numbers are:
a = 2√69 and b = 18 – 2√69
The quadratic equation with these roots is:
(x – 2√69)(x – (18 – 2√69)) = 0
Expanding this equation, we get:
x2 – 20x + 16 = 0
Hence, the correct answer is option D.
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Community Answer
Let two numbers have arithmetic mean 9 and geometric mean 4. Then thes...
Let numbers be x,y now average mean=(x+y)/2 =9 therefore x+y=18 and y=18-x. --------------------(equation 1) geometric mean=√xy=4 xy=16 substituting value of y from equation 1 we get x(18-x)=16 18x-x^2=16 x^2-18x+16 therefore the solution of this equation will give us the numbers hence option d is correct
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Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equationa)x2 + 18x + 16 = 0b)x2 – 18x – 16 = 0c)x2 + 18x – 16 = 0d)x2 – 18x + 16 = 0Correct answer is option 'D'. Can you explain this answer?
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Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equationa)x2 + 18x + 16 = 0b)x2 – 18x – 16 = 0c)x2 + 18x – 16 = 0d)x2 – 18x + 16 = 0Correct answer is option 'D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equationa)x2 + 18x + 16 = 0b)x2 – 18x – 16 = 0c)x2 + 18x – 16 = 0d)x2 – 18x + 16 = 0Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equationa)x2 + 18x + 16 = 0b)x2 – 18x – 16 = 0c)x2 + 18x – 16 = 0d)x2 – 18x + 16 = 0Correct answer is option 'D'. Can you explain this answer?.
Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equationa)x2 + 18x + 16 = 0b)x2 – 18x – 16 = 0c)x2 + 18x – 16 = 0d)x2 – 18x + 16 = 0Correct answer is option 'D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equationa)x2 + 18x + 16 = 0b)x2 – 18x – 16 = 0c)x2 + 18x – 16 = 0d)x2 – 18x + 16 = 0Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equationa)x2 + 18x + 16 = 0b)x2 – 18x – 16 = 0c)x2 + 18x – 16 = 0d)x2 – 18x + 16 = 0Correct answer is option 'D'. Can you explain this answer?.
Solutions for Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equationa)x2 + 18x + 16 = 0b)x2 – 18x – 16 = 0c)x2 + 18x – 16 = 0d)x2 – 18x + 16 = 0Correct answer is option 'D'. Can you explain this answer? in English & in Hindi are available as part of our courses for JEE.
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Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equationa)x2 + 18x + 16 = 0b)x2 – 18x – 16 = 0c)x2 + 18x – 16 = 0d)x2 – 18x + 16 = 0Correct answer is option 'D'. Can you explain this answer?, a detailed solution for Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equationa)x2 + 18x + 16 = 0b)x2 – 18x – 16 = 0c)x2 + 18x – 16 = 0d)x2 – 18x + 16 = 0Correct answer is option 'D'. Can you explain this answer? has been provided alongside types of Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equationa)x2 + 18x + 16 = 0b)x2 – 18x – 16 = 0c)x2 + 18x – 16 = 0d)x2 – 18x + 16 = 0Correct answer is option 'D'. Can you explain this answer? theory, EduRev gives you an
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