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Question

For the quadratic equation ax2+bx+c=0; a,b,cR,
which of the following is/are true ? (where Δ=b24ac,)

A
roots are negative, if all three of a,b,c have the same sign and Δ>0
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B
x=0 is a repeated root, if b=c=0
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C
if c=0, then atleast one root is zero.
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D
roots are negative, if a,c have the same sign but b is of opposite sign and Δ>0
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Solution

The correct option is C if c=0, then atleast one root is zero.
ax2+bx+c=0; a,b,cR
If b=c=0
ax2=0
x=0,0

If c=0, then
ax2+bx=0
(ax+b)x=0
x=0,ba

For real roots, Δ=b24ac0
Let roots are m and n
m+n=ba, mn=ca
Since, roots are negative,
Therefore, both ba and ca are positive.
ba+vea,b+ve or a,bve
ca+vea,c+ve or a,cve
Either a,b,c+ve or a,b,cve

Hence, if roots are negative, then all three of a,b,c have the same sign.

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