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Question

# Consider f(x)=ax2+bx+c with a>0, If both roots of the quadratic equation are greater than any constant k. The necessary and sufficient condition for this are :

A
D0
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B
b2a<k
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C
f(k)>0
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D
b2a>k
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Solution

## The correct option is D −b2a>k Let, α,β be the roots of f(x)=ax2+bx+c When both roots of f(x) are greater than k. From graph, D>0 when roots are distinct, and D=0 when roots are equal. So, we can express it as D≥0 x-coordinates of vertex, i.e, −b2a>k And f(k)>0 So, required conditions are (i) D≥0 (ii) −b2a>k and (iii) f(k)>0

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