Let p and q be the roots of the quadratic equation x2– (α – 2)x– α –1 = 0.…
2025
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.
2
- B.
3
- C.
4
- D.
5
Attempted by 2 students.
Show answer & explanation
Correct answer: D
Given the quadratic equation: x^2 - (alpha - 2)x - alpha - 1 = 0. The roots are p and q.
Step-by-Step Solution
Use the relation between roots and coefficients: For x^2 - Sx + P = 0, the sum of roots p + q = (alpha - 2) and the product of roots pq = -(alpha + 1).
Express p^2 + q^2 in terms of alpha: Using the identity p^2 + q^2 = (p + q)^2 - 2pq: p^2 + q^2 = (alpha - 2)^2 - 2(-(alpha + 1)) p^2 + q^2 = (alpha^2 - 4alpha + 4) + (2alpha + 2) p^2 + q^2 = alpha^2 - 2*alpha + 6
Find the minimum value: The expression alpha^2 - 2alpha + 6 represents a parabola opening upwards. Its minimum value occurs at alpha = -b / (2a). For alpha^2 - 2*alpha + 6, the vertex is at alpha = -(-2) / (2 * 1) = 1. Substituting alpha = 1 into the expression: 1^2 - 2(1) + 6 = 1 - 2 + 6 = 5.