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Let $a, b$ be non-zero real numbers. Which of the following statements about the quadratic equation
$a x^{2}+(a+b) x+b=0$
is neccesarily true?
$(\mathrm{I})$ It has at least one negative root
$(\mathrm{II})$ It has at least one positive root.
$(\mathrm{III})$ Both its roots are real.
Options:
$a x^{2}+(a+b) x+b=0$
is neccesarily true?
$(\mathrm{I})$ It has at least one negative root
$(\mathrm{II})$ It has at least one positive root.
$(\mathrm{III})$ Both its roots are real.
Solution:
1245 Upvotes
Verified Answer
The correct answer is:
$(\mathrm{I})$ and $(\mathrm{III})$ only
$a x^{2}+(a+b) x+b=0$
$(x+1)(a x+b)=0$ roots are $-1, \frac{-b}{a}$
$(x+1)(a x+b)=0$ roots are $-1, \frac{-b}{a}$
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