Question

if alpha and beta are the zeros of the polynomial x2 - 7x + k where alpha - beta = 5,find the value of k and also the values of alpha and beta.

Answer 1

Answer:

Step-by-step explanation:

refer to the photos

Answer 2

$Compare \: given \: polynomial \\x^{2} - 7x + k\: with \:ax^{2} + bx + c , we \:get$

$a = 1 , \: b = -7 \: and \: c = k$

$\alpha \: and \:\beta \: are \: zeroes \: of \\given \: polynomial$

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

$\implies \alpha + \beta = \frac{-(-7)}{1} \\= 7 \: --(1)$

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

$\implies \alpha \beta = \frac{k}{1} \\= k \: --(2)$

$But , \: \alpha - \beta = 5 \: ---(3) \: (given)$

$Add \: equations \: (1) \:and \: (3) ,\: we \:get$

$\implies 2\beta = 12$

$\implies \beta = \frac{12}{2} = 6$

$Put \: \beta = 6 \: in \: Equation \:(1) ,we \:get$

$\implies \alpha + 6 = 7$

$\implies \alpha = 7 - 6$

$\implies \alpha = 1$

$Put \: \alpha = 1 \: and \: \beta = 6 , we \:get$

$\implies 1 \times 6 = k$

$\implies k = 6$

Therefore.,

$\red { Value \: of \: k } \green {= 6}$

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