CAT 2017 Slot 2 — QA Question 24
Forming a Quadratic Equation and Relation between roots and coefficientsEasy
The minimum possible value of the squares of the roots of the equation: x2 + (a + 3)x – (a + 5) = 0 is
Answer & solution
- A
1
- B
2
3
- D
4
Solution
In the given equation, let α and β be the 2 roots of the equation Now α + β = - (a + 3)
Also α × β = - (a – 5)
(α + β)2 = α2 + β2 + 2αβ
⇒ (a + 3)2 = α2 + β2 - 2(a + 5)
âââââââ⇒ a2 + 6a + 9 = α2 + β2 - 2a – 10
⇒ a2 + 8a + 19 = α2 + β2
⇒ (a2 + 8a + 16) + 3 = α2 + β2
⇒ α2 + β2 = (a + 4)2 + 3
Now (a + 4)2 ≥ 0
So minimum value of (a + 4)2 will be 0 when a = -4
At a = -4, the of squares of the roots of the equation i.e., α2 + β2 will be 3.
Hence, option (c).