Question

if alpha and beta are zeroes of polynomial f(x)=x2-px+q then find values of-
(i)alpha^2+beta^2
(i)1/alpha+1/beta

Answer 1

hope this helps and good luck for exam

Answer 2

Answer:

1)$\alpha^{2}+\beta^{2}$=p²-2q

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

=$\frac{p}{q}$

Explanation:

It is given that ,

$\alpha\: and \beta \:$ are two zeroes of quadratic polynomial f(x)=x²-px+q

Compare f(x) with ax²+bx+c , we get

a = 1 , b = -p, c = q

Now,

i ) Sum of the zeroes = $\frac{-b}{a}$

$\implies \alpha + \beta = \frac{-(-p)}{1}$

$\implies \alpha + \beta = p$ -----(1)

ii) Product of the zeroes = $\frac{c}{a}$

$\implies \alpha\beta=\frac{q}{1}$

$\implies \alpha\beta=q$-----(2)

Now ,

iii ) $\alpha^{2}+\beta^{2}=\left(\alpha+\beta\right)^{2}-2\alpha\beta$

= $p^{2}-2q$

/* From (1) and (2) */

iv) $\frac{1}{\alpha}+\frac{1}{\beta}$

= $\frac{(\alpha+\beta)}{\alpha\beta}$

= $\frac{p}{q}$

/* from (1) and (2) */

Therefore,

1)$\alpha^{2}+\beta^{2}$=p²-2q

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

=$\frac{p}{q}$

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