Question

find a condition that zeroes of the polynomial
$p(x) = a {x}^{2} + bx + c$
are reciprocal of each other.​

*please answer the question with explanation.*

Answer 1

$\huge{ \green{ \mathfrak{ \underline{Question}}}}$

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Find a condition that zeroes of the polynomial

$p(x) = a {x}^{2} + bx + c$ are reciprocal of each other.

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$\huge{ \green{ \mathfrak{ \underline{Answer}}}}$

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☆$\boxed{Degree \:of \:polynomial \:is\: 2,\: so \:it \:has \:2 \:zeroes}$

$\boxed{\red{\bold{Let\: the\: zeroes\:are:}}}$

$\alpha , \beta$

$Using \: the \:given \:relation , \: \beta = \displaystyle{\frac{1}{\alpha}}$

$\boxed{\red{\bold{Relation\:of\:zeroes\: and \: coefficients:}}}$

$Product\:of \:zeroes= \displaystyle{\frac{Constant\: term}{Coefficient \: of \: x^2}}$

$\implies \alpha \times \beta = \displaystyle{\frac{c}{a}}$

$\implies \alpha \times \displaystyle{\frac{1}{\alpha}} = \frac{c}{a}\; \; \; \left( \because \beta = \displaystyle{\frac{1}{\alpha}} \right)$

$\implies 1 = \displaystyle{\frac{c}{a}}$

$\implies c = a$

☆Hence one condition can be that the constant term and coefficient of $x^2$ are equal , i.e., $a=c$

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