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Question

If the roots of the quadratic equation x2px+q=0 are real and differ by a quantity less than 1, then

A
None of these
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B
q>p24
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C
p214<q<1+p24
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D
p214<q<p24
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Solution

The correct option is C p214<q<1+p24
Given equation
x2px+q=0
Comparing it with standard equation
ax2+bx+c=0
a=1,b=p,c=q
Given condition is roots of the given quadratic equation differs by a quantity less than 1.
Mathematical representstion of this condition is
|αβ|<1
We know that
(αβ)2=Da2

Now,
D|a|<1
(p)24×1×q1<1
p24q<1
Squaring the inequality
|p24q|<1
1<p24q<1
1p2<4q<1p2
1p24<q<1p24
p214<q<1+p24

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