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If α , β are the roots of the quadratic equation x2 − (a − 2) x − (a + 1) = 0 , where a is a variable, then the least value of α2 + β2 is
- a)3
- b)5
- c)7
- d)None
Correct answer is option 'B'. Can you explain this answer?
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If α , β are the roots of the quadratic equation x2 −...
Given information:
- Roots of the quadratic equation: α and β
- Quadratic equation: x^2 - (a - 2)x - (a + 1) = 0
Objective:
To find the least value of α^2 + β^2
Solution:
Sum of roots:
From the quadratic equation, we know that the sum of the roots is given by:
α + β = a - 2
Product of roots:
Similarly, the product of the roots is given by:
αβ = -(a + 1)
Value of α^2 + β^2:
We know that:
α^2 + β^2 = (α + β)^2 - 2αβ
Substitute the values of sum and product of roots:
α^2 + β^2 = (a - 2)^2 - 2(-(a + 1))
α^2 + β^2 = a^2 - 4a + 4 + 2a + 2
α^2 + β^2 = a^2 - 2a + 6
Least value of α^2 + β^2:
To find the least value, take the derivative of the expression with respect to 'a' and set it to zero:
d(α^2 + β^2)/da = 2a - 2 = 0
a = 1
Substitute 'a = 1' back into the expression:
α^2 + β^2 = 1^2 - 2(1) + 6
α^2 + β^2 = 5
Therefore, the least value of α^2 + β^2 is 5, which corresponds to option 'B'.
- Roots of the quadratic equation: α and β
- Quadratic equation: x^2 - (a - 2)x - (a + 1) = 0
Objective:
To find the least value of α^2 + β^2
Solution:
Sum of roots:
From the quadratic equation, we know that the sum of the roots is given by:
α + β = a - 2
Product of roots:
Similarly, the product of the roots is given by:
αβ = -(a + 1)
Value of α^2 + β^2:
We know that:
α^2 + β^2 = (α + β)^2 - 2αβ
Substitute the values of sum and product of roots:
α^2 + β^2 = (a - 2)^2 - 2(-(a + 1))
α^2 + β^2 = a^2 - 4a + 4 + 2a + 2
α^2 + β^2 = a^2 - 2a + 6
Least value of α^2 + β^2:
To find the least value, take the derivative of the expression with respect to 'a' and set it to zero:
d(α^2 + β^2)/da = 2a - 2 = 0
a = 1
Substitute 'a = 1' back into the expression:
α^2 + β^2 = 1^2 - 2(1) + 6
α^2 + β^2 = 5
Therefore, the least value of α^2 + β^2 is 5, which corresponds to option 'B'.
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If α , β are the roots of the quadratic equation x2 − (a − 2) x − (a + 1) = 0 , where a is a variable, then the least value of α2 + β2 isa)3b)5c)7d)NoneCorrect answer is option 'B'. Can you explain this answer?
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