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Nature of Roots

If α, β a...

Question

If $α$ , $β$ are the zeroes of the polynomial $f (x) = x^{2} - 5 x + k$ such that $α - β = 1$ , find the value of $k$ .

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Solution

$H e r e t h e g i v e n e q u a t i o n i s x^{2} - 5 x + k$
$I f w e c o n s i d e r α a n d β a s r o o t s o f t h e e q u a t i o n, t h e n$
$S u m o f z e r o s, α + β = - \frac{b}{a} = - \frac{(- 5)}{1} = 5 - - - - (1)$
$A l s o i t i s g i v e n α - β = 1 - - - - - (2)$
$A d d i n g (1) a n d (2) w e g e t$
$2 α = 6$
$α = 3$
$P u t α = 3 i n (1), 3 + β = 5$
$β = 2$
$N o w p r o d u c t o f z e r o s, α β = \frac{c}{a} = k$
$U s i n g v a l u e o f α a n d β w e g e t$
$α β = k$
$k = 3 \cdot 2 = 6$
$T h u s k = 6$

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