Question

If alpha and beta are the zeros of the polynomial x2-6x+b prove find the value of b if 3 alpha +2 beta=20

Answer 1

$p(x) = {x}^{2} - 6x + b$
here,

a= 1, b= -6 and c= b

we know that,

sum of zeroes = -b/a

$\alpha + \beta = 6$
__________________________(1)

Also,

Product of zeroes= c/a

$\alpha \beta = b$
__________________________(2)
given:

$3 \alpha + 2 \beta = 20$
__________________________(3)


multiply 2 to equation(1) and subtract (1) from (3).


$3 \alpha + 2 \beta = 20 \\ \\ 2 \alpha + 2 \beta = 12 \: \: \: (on \: subtracting) \\ ........................ \\ \: \: \: \alpha \: \: \: \: = \: \: 8 \\ .......................... \\ \\ put \: this \: in \: (1) \\ \\ 8 + \beta = 6 \\ \\ \beta = - 2$
$substitue \: the \: values \: of \\ \: \alpha \: and \: \beta \: in \: (2)$
$\alpha \beta = b \\ \\ - 2 \times 8 = b \\ \\ - 16 = b$

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