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If α and β are the zeroes of the quadratic polynomial p(s) =3s2-6s +4, find the value of α/β+ β/α + 2 (1/α+ 1/β) + 3 αβ?
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If α and β are the zeroes of the quadratic polynomial p(s) =3s2-6s +4,...
Finding the value of α/β + β/α + 2(1/α + 1/β) + 3αβ
To find the value of the expression given in the question, we first need to understand the relationship between the zeroes of a quadratic polynomial and the coefficients of the polynomial.
Relationship between zeroes and coefficients:
For a quadratic polynomial of the form p(s) = as^2 + bs + c, the sum of the zeroes (α + β) is given by -b/a and the product of the zeroes (αβ) is given by c/a.
Given quadratic polynomial:
In this case, the quadratic polynomial is p(s) = 3s^2 - 6s + 4.
Using the relationship mentioned above, we can find the sum and product of the zeroes of the polynomial:
α + β = -(-6)/3 = 2
αβ = 4/3
Expression to evaluate:
The expression we need to evaluate is α/β + β/α + 2(1/α + 1/β) + 3αβ.
Substitute the values of α + β and αβ into the expression:
α/β + β/α + 2(1/α + 1/β) + 3αβ
= (α^2 + β^2)/αβ + 2(α + β)/αβ + 3αβ
= ((α + β)^2 - 2αβ)/αβ + 2(α + β)/αβ + 3αβ
= (2^2 - 2(4/3))/(4/3) + 2(2)/(4/3) + 3(4/3)
= (4 - 8/3)/(4/3) + 4/(4/3) + 4
= (12/3 - 8/3)/(4/3) + 12/3 + 12/3
= 4/3 * 4/3 + 12/3 + 12/3
= 16/9 + 12/3 + 12/3
= 16/9 + 4 + 4
= 16/9 + 8
= (16 + 72)/9
= 88/9
Therefore, the value of the expression α/β + β/α + 2(1/α + 1/β) + 3αβ is 88/9.
To find the value of the expression given in the question, we first need to understand the relationship between the zeroes of a quadratic polynomial and the coefficients of the polynomial.
Relationship between zeroes and coefficients:
For a quadratic polynomial of the form p(s) = as^2 + bs + c, the sum of the zeroes (α + β) is given by -b/a and the product of the zeroes (αβ) is given by c/a.
Given quadratic polynomial:
In this case, the quadratic polynomial is p(s) = 3s^2 - 6s + 4.
Using the relationship mentioned above, we can find the sum and product of the zeroes of the polynomial:
α + β = -(-6)/3 = 2
αβ = 4/3
Expression to evaluate:
The expression we need to evaluate is α/β + β/α + 2(1/α + 1/β) + 3αβ.
Substitute the values of α + β and αβ into the expression:
α/β + β/α + 2(1/α + 1/β) + 3αβ
= (α^2 + β^2)/αβ + 2(α + β)/αβ + 3αβ
= ((α + β)^2 - 2αβ)/αβ + 2(α + β)/αβ + 3αβ
= (2^2 - 2(4/3))/(4/3) + 2(2)/(4/3) + 3(4/3)
= (4 - 8/3)/(4/3) + 4/(4/3) + 4
= (12/3 - 8/3)/(4/3) + 12/3 + 12/3
= 4/3 * 4/3 + 12/3 + 12/3
= 16/9 + 12/3 + 12/3
= 16/9 + 4 + 4
= 16/9 + 8
= (16 + 72)/9
= 88/9
Therefore, the value of the expression α/β + β/α + 2(1/α + 1/β) + 3αβ is 88/9.
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If α and β are the zeroes of the quadratic polynomial p(s) =3s2-6s +4, find the value of α/β+ β/α + 2 (1/α+ 1/β) + 3 αβ? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about If α and β are the zeroes of the quadratic polynomial p(s) =3s2-6s +4, find the value of α/β+ β/α + 2 (1/α+ 1/β) + 3 αβ? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If α and β are the zeroes of the quadratic polynomial p(s) =3s2-6s +4, find the value of α/β+ β/α + 2 (1/α+ 1/β) + 3 αβ?.
If α and β are the zeroes of the quadratic polynomial p(s) =3s2-6s +4, find the value of α/β+ β/α + 2 (1/α+ 1/β) + 3 αβ? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about If α and β are the zeroes of the quadratic polynomial p(s) =3s2-6s +4, find the value of α/β+ β/α + 2 (1/α+ 1/β) + 3 αβ? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If α and β are the zeroes of the quadratic polynomial p(s) =3s2-6s +4, find the value of α/β+ β/α + 2 (1/α+ 1/β) + 3 αβ?.
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