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Question

# Set of values of a for which both the roots of the quadratic polynomial f(x)=ax2+(a−3)x+1 lie on one side of the y−axis is

A
[0,1)[3,)
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B
[0,1][9,)
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C
(0,1)(3,)
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D
(0,1][9,)
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Solution

## The correct option is D (0,1]∪[9,∞)For the given polynomial, a≠0 ∵ Roots are only on one side, ∴ Both roots either be +ve or −ve For both the roots to be real, D≥0⇒(a−3)2−4a≥0⇒a2−10a+9≥0⇒a∈(−∞,1]∪[9,∞)−{0} ⋯(1) Now, Case 1: Both roots are positive Product and sum of the roots both will be >0 ∴−a−3a>0, 1a>0⇒a−3a<0, a∈(0,∞)⇒a∈(0,3), a∈(0,∞)⇒a∈(0,3) ⋯(2) Case 2: Both roots are negative Product of the root will be positive and sum will be negative. ∴1a>0, −a−3a<0⇒a∈(0,∞), a−3a>0⇒a∈(3,∞) ⋯(3) From above equations, a∈(0,1]∪[9,∞)

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