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Question:
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Consider the following:
1.$\quad \alpha+\beta+\alpha \beta>0$
2.$\quad \alpha^{2} \beta+\beta^{2} \alpha>0$
Which of the above is/are correct?
Options:
1.$\quad \alpha+\beta+\alpha \beta>0$
2.$\quad \alpha^{2} \beta+\beta^{2} \alpha>0$
Which of the above is/are correct?
Solution:
1209 Upvotes
Verified Answer
The correct answer is:
2 only
Sum of roots $=\alpha+\beta=-b$ Multiplication of roots $=\alpha \beta=c$ Hence $\begin{aligned} \alpha+\beta+\alpha \beta=-b+c & \alpha^{2} \beta+\beta^{2}=& \alpha \beta(\alpha+\beta) \\ &=-b c \end{aligned}$
$\because \mathrm{b}>0 \& \mathrm{c} < 0$
$\therefore-\mathrm{b}+\mathrm{c} < 0 \&-\mathrm{bc}>0$
$\because \mathrm{b}>0 \& \mathrm{c} < 0$
$\therefore-\mathrm{b}+\mathrm{c} < 0 \&-\mathrm{bc}>0$
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