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If α, β are the roots of the equations x2 – 2x– 1 = 0, then what is the value of α2 β–2+ α –2 β2
- a)–2
- b)0
- c)30
- d)34
Correct answer is option 'D'. Can you explain this answer?
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If α, β are the roots of the equations x2 – 2x–...
Solution:
Given:
The roots of the quadratic equation x^2 - 2x - 1 = 0 are α and β.
To Find:
We need to find the value of α^2β - 2 + α - 2β^2.
Explanation:
From the given quadratic equation x^2 - 2x - 1 = 0, we know that the sum of the roots α + β = 2 and the product of the roots αβ = -1.
Now, we need to find the value of α^2β - 2 + α - 2β^2.
α^2β - 2 + α - 2β^2
= α(αβ) - 2 + α - 2(β^2)
= α(-1) - 2 + α - 2(β^2)
= -α + α - 2 - 2(β^2)
= -2 - 2(β^2)
= -2(1 + β^2)
Now, we know that (α + β)^2 = α^2 + β^2 + 2αβ = 4 (from the sum of roots)
Therefore, α^2 + β^2 = 4 - 2αβ
α^2 + β^2 = 4 - 2*(-1)
α^2 + β^2 = 4 + 2
α^2 + β^2 = 6
Now, substituting the value of α^2 + β^2 = 6 into the expression -2(1 + β^2), we get:
-2(1 + β^2) = -2(1 + 1)
-2(1 + β^2) = -2(2)
-2(1 + β^2) = -4
Therefore, the value of α^2β - 2 + α - 2β^2 is -4.
So, the correct answer is option D which is -4.
Given:
The roots of the quadratic equation x^2 - 2x - 1 = 0 are α and β.
To Find:
We need to find the value of α^2β - 2 + α - 2β^2.
Explanation:
From the given quadratic equation x^2 - 2x - 1 = 0, we know that the sum of the roots α + β = 2 and the product of the roots αβ = -1.
Now, we need to find the value of α^2β - 2 + α - 2β^2.
α^2β - 2 + α - 2β^2
= α(αβ) - 2 + α - 2(β^2)
= α(-1) - 2 + α - 2(β^2)
= -α + α - 2 - 2(β^2)
= -2 - 2(β^2)
= -2(1 + β^2)
Now, we know that (α + β)^2 = α^2 + β^2 + 2αβ = 4 (from the sum of roots)
Therefore, α^2 + β^2 = 4 - 2αβ
α^2 + β^2 = 4 - 2*(-1)
α^2 + β^2 = 4 + 2
α^2 + β^2 = 6
Now, substituting the value of α^2 + β^2 = 6 into the expression -2(1 + β^2), we get:
-2(1 + β^2) = -2(1 + 1)
-2(1 + β^2) = -2(2)
-2(1 + β^2) = -4
Therefore, the value of α^2β - 2 + α - 2β^2 is -4.
So, the correct answer is option D which is -4.
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If α, β are the roots of the equations x2 – 2x– 1 = 0, then what is the value of α2 β–2+ α –2 β2a)–2b)0c)30d)34Correct answer is option 'D'. Can you explain this answer?
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If α, β are the roots of the equations x2 – 2x– 1 = 0, then what is the value of α2 β–2+ α –2 β2a)–2b)0c)30d)34Correct 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 If α, β are the roots of the equations x2 – 2x– 1 = 0, then what is the value of α2 β–2+ α –2 β2a)–2b)0c)30d)34Correct 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 If α, β are the roots of the equations x2 – 2x– 1 = 0, then what is the value of α2 β–2+ α –2 β2a)–2b)0c)30d)34Correct answer is option 'D'. Can you explain this answer?.
If α, β are the roots of the equations x2 – 2x– 1 = 0, then what is the value of α2 β–2+ α –2 β2a)–2b)0c)30d)34Correct 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 If α, β are the roots of the equations x2 – 2x– 1 = 0, then what is the value of α2 β–2+ α –2 β2a)–2b)0c)30d)34Correct 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 If α, β are the roots of the equations x2 – 2x– 1 = 0, then what is the value of α2 β–2+ α –2 β2a)–2b)0c)30d)34Correct answer is option 'D'. Can you explain this answer?.
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