Question

$if \: \alpha\:and\: \beta \: are \: the \: zeroes \: of \: the \: polynomial \: f(x) = {ax}^{2} + bx + c \: \: then \:find \: the \: value \: of \: \frac{1}{ { \alpha }^{2} } + \frac{1}{ { \beta }^{2} }$
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Answer 1

Answer:

$\frac{ {b}^{2} - 2ac}{c {}^{2} }$

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Step-by-step explanation:

Given,

$\alpha\:and\: \beta \: are \: the \: zeroes \: of \: the \: polynomial \: f(x) = {ax}^{2} + bx + c \: \:$

f(x) = ax² + bc + c

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To find,

$\frac{1}{ \alpha {}^{2} } + \frac{1}{ \beta {}^{2} }$

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Solution,

$\frac{1}{ \alpha {}^{2} } + \frac{1}{ \beta {}^{2} } \\ \\ = \frac{ \alpha {}^{2} + \beta {}^{2} }{ \alpha {}^{2} \beta {}^{2} } \\ \\ \\ = \frac{( \alpha + \beta ) {}^{2} - 2 \alpha \beta }{( \alpha \beta ) {}^{2} } \\ \\ = \frac{ \frac{b {}^{2} }{a {}^{2} } - \frac{2c}{a} }{ \frac{c {}^{2} }{a {}^{2} } } \\ \\ = \frac{b {}^{2} - 2ac}{c {}^{2} }$

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Sum of zeroes in a quadratic polynomial

α + β = -b/a = (coefficient of x)/(coefficient of x²)

Product of zeroes in a quadratic polynomial

αβ = c/a = constant term/coefficient of x²