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If α, β are the roots of the quadratic equation ax2 + bx + c = 0 and 3b2 = 16ac then
- a)α = 4β or β = 4α
- b)α = –4β or β = –4α
- c)α = 3β or β = 3α
- d)α = –3β or β = –3α
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
If α, β are the roots of the quadratic equation ax2 + bx + ...
3b2 = 16ac
Most Upvoted Answer
If α, β are the roots of the quadratic equation ax2 + bx + ...
Alpha * beeta = c/a equation (l)
alpha + beeta = -b/a equation (1)
doing square both sides before multiplying by (3a^2) both sides in equation (1) :-
3a^2(alpha + beta )^2 = 3 b^2
so,
3a^2(alpha square + beeta square + 2alpha beeta) = 16ac
or,
(alpha square + beeta square + 2c/a ) = (16/3)c/a
alpha square + beeta square = ( 16c/3a ) - (2c/a)
alpha square + beeta square = (16c - 6c)/3a
alpha square + beeta square = (10/3)(alpha * beeta) (using equation (l))
so,
3alpha square + 3 beeta square - 2 x 5 alpha * beeta = 0
let :- alpha = t , beeta = b
3t^2 - 10tb + 3b^2 = 0
t = (10b + √(100b^2 - 36b^2))/6
or,
t = ( 10b + 8b) /6
= 18b/6. = 3b
again similarly,
t = (10b - 8b)/6 = b/3
so,
3t = b
hence:-
alpha = 3 beeta
and 3alpha = beeta
thus we can say that the option (c) is the correct option
alpha + beeta = -b/a equation (1)
doing square both sides before multiplying by (3a^2) both sides in equation (1) :-
3a^2(alpha + beta )^2 = 3 b^2
so,
3a^2(alpha square + beeta square + 2alpha beeta) = 16ac
or,
(alpha square + beeta square + 2c/a ) = (16/3)c/a
alpha square + beeta square = ( 16c/3a ) - (2c/a)
alpha square + beeta square = (16c - 6c)/3a
alpha square + beeta square = (10/3)(alpha * beeta) (using equation (l))
so,
3alpha square + 3 beeta square - 2 x 5 alpha * beeta = 0
let :- alpha = t , beeta = b
3t^2 - 10tb + 3b^2 = 0
t = (10b + √(100b^2 - 36b^2))/6
or,
t = ( 10b + 8b) /6
= 18b/6. = 3b
again similarly,
t = (10b - 8b)/6 = b/3
so,
3t = b
hence:-
alpha = 3 beeta
and 3alpha = beeta
thus we can say that the option (c) is the correct option
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Community Answer
If α, β are the roots of the quadratic equation ax2 + bx + ...
Understanding the Given Condition
The condition provided is 3b^2 = 16ac. We can interpret this in terms of the roots of the quadratic equation ax^2 + bx + c = 0, where α and β represent the roots.
Relationship Between Roots and Coefficients
From Vieta's formulas, we know:
- α + β = -b/a
- αβ = c/a
Using these relationships, we can derive a connection between the roots.
Substituting into the Condition
We start with the condition 3b^2 = 16ac. Replacing b with -a(α + β) and c with a(αβ), we get:
3(-a(α + β))^2 = 16a(a(αβ))
This simplifies to:
3a^2(α + β)^2 = 16a^2(αβ)
Cancelling a^2 (assuming a ≠ 0):
3(α + β)^2 = 16(αβ)
Exploring the Roots
Dividing the whole equation by (αβ):
3(α + β)^2/(αβ) = 16
This expression leads to a specific relationship between α and β.
Implication of the Condition
To find the relationship between α and β, we can check possible ratios. After some algebraic manipulation, we find that:
α = 3β or β = 3α
This satisfies our condition, confirming that if the given quadratic satisfies 3b^2 = 16ac, then the roots are indeed in the ratio of 3:1.
Conclusion
Thus, the correct answer is:
- Option C: α = 3β or β = 3α.
This relationship shows how the roots are directly proportional, fulfilling the given quadratic condition.
The condition provided is 3b^2 = 16ac. We can interpret this in terms of the roots of the quadratic equation ax^2 + bx + c = 0, where α and β represent the roots.
Relationship Between Roots and Coefficients
From Vieta's formulas, we know:
- α + β = -b/a
- αβ = c/a
Using these relationships, we can derive a connection between the roots.
Substituting into the Condition
We start with the condition 3b^2 = 16ac. Replacing b with -a(α + β) and c with a(αβ), we get:
3(-a(α + β))^2 = 16a(a(αβ))
This simplifies to:
3a^2(α + β)^2 = 16a^2(αβ)
Cancelling a^2 (assuming a ≠ 0):
3(α + β)^2 = 16(αβ)
Exploring the Roots
Dividing the whole equation by (αβ):
3(α + β)^2/(αβ) = 16
This expression leads to a specific relationship between α and β.
Implication of the Condition
To find the relationship between α and β, we can check possible ratios. After some algebraic manipulation, we find that:
α = 3β or β = 3α
This satisfies our condition, confirming that if the given quadratic satisfies 3b^2 = 16ac, then the roots are indeed in the ratio of 3:1.
Conclusion
Thus, the correct answer is:
- Option C: α = 3β or β = 3α.
This relationship shows how the roots are directly proportional, fulfilling the given quadratic condition.
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If α, β are the roots of the quadratic equation ax2 + bx + c = 0 and 3b2 = 16ac thena)α = 4β or β = 4αb)α = –4β or β = –4αc)α = 3β or β = 3αd)α = –3β or β = –3αCorrect answer is option 'C'. Can you explain this answer?
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If α, β are the roots of the quadratic equation ax2 + bx + c = 0 and 3b2 = 16ac thena)α = 4β or β = 4αb)α = –4β or β = –4αc)α = 3β or β = 3αd)α = –3β or β = –3αCorrect answer is option 'C'. 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 quadratic equation ax2 + bx + c = 0 and 3b2 = 16ac thena)α = 4β or β = 4αb)α = –4β or β = –4αc)α = 3β or β = 3αd)α = –3β or β = –3αCorrect answer is option 'C'. 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 quadratic equation ax2 + bx + c = 0 and 3b2 = 16ac thena)α = 4β or β = 4αb)α = –4β or β = –4αc)α = 3β or β = 3αd)α = –3β or β = –3αCorrect answer is option 'C'. Can you explain this answer?.
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