Question

if alpha & beta are the zeros of the quadratic polynomial f(x) =x^2+px+q then find the values of (1) alpha ^2 + beta ^2 (2) 1/alpha + 1/ beta

Answer 1

[Hello]

Topic:- Polynomial
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Equation:- x²+PX+q

a+b=-b/a=)

-p/1=-p

Now..

ab=c/a=)q/1

ACCORDING TO THE QUESTION;

a²+b²

=)(a+b)²-2ab

=)(-p)²-2.q

=)p²-2q

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1/a+1/b

_b+a_____
ab

=}a+b/ab

=}-p/q

=)-p/q

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Answer 2

$Given - \alpha \: and \: \beta \: are \: zeroes \: of \\ polynomial \: {x }^{2} + px + q$
a = 1
b = p
c = q

We know that, sum of zeroes = $\frac{-b}{a}$

And, Product of zeroes = $\frac{c}{a}$

=> $\alpha + \beta = - p \\ \alpha \beta = q$

We need to find,
i) ${ \alpha }^{2} + { \beta }^{2}$

Using the identity,
$( {a}^{2} + {b}^{2} = {(a + b)}^{2} - 2ab )$
$= > \: { (- p)}^{2} + 2(q) \\ = > {p}^{2} + 2q$

And, ii)$\frac{1}{ \alpha } + \frac{1}{ \beta }$

=>$\frac{ \alpha + \beta }{ \alpha \beta }$
=> $\frac{ - p}{q}$