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If alpha and beta are zeroes of the polynomial kx^2 4x 4 such that alpha square beta square=24,find k?
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If alpha and beta are zeroes of the polynomial kx^2 4x 4 such that alp...
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If alpha and beta are zeroes of the polynomial kx^2 4x 4 such that alp...
Explanation:
Given polynomial is kx^2 +4x+4=0. Let α and β be the roots of this polynomial.
Then, we have:
α+β=-4/k
αβ=1
α^2β^2=24
Solution:
We need to find the value of k.
From the equation α+β=-4/k, we have:
α+β=-4/k
αβ=1
Squaring both sides of α+β=-4/k, we get:
(α+β)^2=16/k^2
α^2+2αβ+β^2=16/k^2
α^2+β^2+2=16/k^2
α^2+β^2=16/k^2-2
Now, using the equation α^2β^2=24, we have:
α^2β^2=24
(α^2+β^2)^2-2α^2β^2=24
(16/k^2-2)^2-2(1)=24
Simplifying the above equation:
256/k^4 - 64/k^2 + 4 - 2 = 24
256/k^4 - 64/k^2 = 22
Dividing by 2:
128/k^4 - 32/k^2 = 11
Multiplying by k^4, we get:
128 - 32k^2 = 11k^4
11k^4 + 32k^2 - 128 = 0
This is a quadratic equation in k^2. Solving this quadratic equation using the quadratic formula:
k^2 = [-32 ± sqrt(32^2 - 4(11)(-128))] / 22
k^2 = [-32 ± sqrt(1696)] / 22
k^2 = [-32 ± 8sqrt(26)] / 11
Since k is positive, we take the positive square root:
k = sqrt{[-32 + 8sqrt(26)] / 11}
Hence, the value of k is:
k = sqrt{[-32 + 8sqrt(26)] / 11}
Therefore, k is approximately equal to 1.532.
Given polynomial is kx^2 +4x+4=0. Let α and β be the roots of this polynomial.
Then, we have:
α+β=-4/k
αβ=1
α^2β^2=24
Solution:
We need to find the value of k.
From the equation α+β=-4/k, we have:
α+β=-4/k
αβ=1
Squaring both sides of α+β=-4/k, we get:
(α+β)^2=16/k^2
α^2+2αβ+β^2=16/k^2
α^2+β^2+2=16/k^2
α^2+β^2=16/k^2-2
Now, using the equation α^2β^2=24, we have:
α^2β^2=24
(α^2+β^2)^2-2α^2β^2=24
(16/k^2-2)^2-2(1)=24
Simplifying the above equation:
256/k^4 - 64/k^2 + 4 - 2 = 24
256/k^4 - 64/k^2 = 22
Dividing by 2:
128/k^4 - 32/k^2 = 11
Multiplying by k^4, we get:
128 - 32k^2 = 11k^4
11k^4 + 32k^2 - 128 = 0
This is a quadratic equation in k^2. Solving this quadratic equation using the quadratic formula:
k^2 = [-32 ± sqrt(32^2 - 4(11)(-128))] / 22
k^2 = [-32 ± sqrt(1696)] / 22
k^2 = [-32 ± 8sqrt(26)] / 11
Since k is positive, we take the positive square root:
k = sqrt{[-32 + 8sqrt(26)] / 11}
Hence, the value of k is:
k = sqrt{[-32 + 8sqrt(26)] / 11}
Therefore, k is approximately equal to 1.532.
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If alpha and beta are zeroes of the polynomial kx^2 4x 4 such that alpha square beta square=24,find k?
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If alpha and beta are zeroes of the polynomial kx^2 4x 4 such that alpha square beta square=24,find k? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If alpha and beta are zeroes of the polynomial kx^2 4x 4 such that alpha square beta square=24,find k? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If alpha and beta are zeroes of the polynomial kx^2 4x 4 such that alpha square beta square=24,find k?.
If alpha and beta are zeroes of the polynomial kx^2 4x 4 such that alpha square beta square=24,find k? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If alpha and beta are zeroes of the polynomial kx^2 4x 4 such that alpha square beta square=24,find k? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If alpha and beta are zeroes of the polynomial kx^2 4x 4 such that alpha square beta square=24,find k?.
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