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Let p and q be the roots of the quadratic equation x2 - (α - 2)x - α - 1 = 0. What is the minimum possible value of p2 + q2?
- a)0
- b)3
- c)4
- d)5
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
Let p and q be the roots of the quadratic equation x2 - (α - 2)x...
Understanding the Quadratic Equation
The given quadratic equation is:
- x² - (α - 2)x - (α + 1) = 0
Using Vieta's formulas, we know that:
- p + q = α - 2 (sum of roots)
- pq = - (α + 1) (product of roots)
Finding p² + q²
To find the minimum possible value of p² + q², we can use the identity:
- p² + q² = (p + q)² - 2pq
Substituting from Vieta's formulas:
- p² + q² = (α - 2)² - 2(- (α + 1))
This simplifies to:
- p² + q² = (α - 2)² + 2(α + 1)
Expanding the Expression
Now, we expand the equation:
- p² + q² = (α² - 4α + 4) + (2α + 2)
- p² + q² = α² - 2α + 6
Finding the Minimum Value
To minimize α² - 2α + 6, we consider it as a quadratic function:
- Minimum occurs at α = -b/2a = 2/2 = 1
Substituting α = 1 back into the expression gives:
- p² + q² = (1)² - 2(1) + 6 = 1 - 2 + 6 = 5
Conclusion
The minimum possible value of p² + q² is:
- 5
Thus, the correct answer is option 'D'.
The given quadratic equation is:
- x² - (α - 2)x - (α + 1) = 0
Using Vieta's formulas, we know that:
- p + q = α - 2 (sum of roots)
- pq = - (α + 1) (product of roots)
Finding p² + q²
To find the minimum possible value of p² + q², we can use the identity:
- p² + q² = (p + q)² - 2pq
Substituting from Vieta's formulas:
- p² + q² = (α - 2)² - 2(- (α + 1))
This simplifies to:
- p² + q² = (α - 2)² + 2(α + 1)
Expanding the Expression
Now, we expand the equation:
- p² + q² = (α² - 4α + 4) + (2α + 2)
- p² + q² = α² - 2α + 6
Finding the Minimum Value
To minimize α² - 2α + 6, we consider it as a quadratic function:
- Minimum occurs at α = -b/2a = 2/2 = 1
Substituting α = 1 back into the expression gives:
- p² + q² = (1)² - 2(1) + 6 = 1 - 2 + 6 = 5
Conclusion
The minimum possible value of p² + q² is:
- 5
Thus, the correct answer is option 'D'.
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Community Answer
Let p and q be the roots of the quadratic equation x2 - (α - 2)x...
Let α be equal to k.
⇒ f(x) = x2 − (k − 2) x − (k + 1) = 0
p and q are the roots
⇒ p + q = k - 2 and pq = -1 - k
We know that (p + q)2 = p2 + q2 + 2pq
⇒ (k − 2)2 = p2 + q2 + 2(−1 − k)
⇒ p2 + q2 = k2 + 4 − 4k + 2 + 2k
⇒ p2 + q2 = k2 − 2k + 6
This is in the form of a quadratic equation.
The coefficient of k2 is positive. Therefore this equation has a minimum value.
We know that the minimum value occurs at x = -b/2a
Here a = 1, b = -2 and c = 6
⇒ Minimum value occurs at k = 2/2 = 1
If we substitute k = 1 in k2 − 2k + 6, we get 1 -2 + 6 = 5.
Hence 5 is the minimum value that p2 + q2 can attain.
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