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Question

# For the quadratic equation ax2+bx+c=0; a,b,c∈R, which of the following is/are true ? (where Δ=b2−4ac,)

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

## The correct options are A x=0 is a repeated root, if b=c=0 B if c=0, then atleast one root is zero. D roots are negative, if all three of a,b,c have the same sign and Δ>0 ax2+bx+c=0; a,b,c∈R 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=b2−4ac≥0 Let roots are m and n m+n=−ba, mn=ca Since, roots are negative, Therefore, both ba and ca are positive. ba→+ve⇒a,b→+ve or a,b→−ve ca→+ve⇒a,c→+ve or a,c→−ve ⇒ Either a,b,c→+ve or a,b,c→−ve Hence, if roots are negative, then all three of a,b,c have the same sign.

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