Question

find the value of k ...N plz.give clear steps

Answer 1

Hey.

As for any quadratic equation of the form

$a {x}^{2} + bx + c = 0 \\ \\ sum \: of \: zeros \: = \frac{ - b}{a} \\ \\ and \: product \: of \: zeros \: = \frac{c}{a}$

writing alpha as a and beta as b

here
$a \: + b \: = \frac{ - 4}{k} \\ \\ and \: \: ab \: = \: \frac{4}{k}$

As

${a}^{2} + {b}^{2} = 24 \\ \\ \\ we \: know \: {(a + b) }^{2} = {a }^{2} + {b}^{2} + 2ab \\ \\ so \: \frac{ {( - 4)}^{2} }{ {k}^{2} } = 24 + 2 \times \frac{4}{k} \\ \\ so \: \frac{16}{ {k}^{2} } = 24 + \frac{8}{k} \\ \\ or \: \frac{16}{ {k}^{2} } = \frac{24k + 8}{k} \\ \\ \: or \: \frac{16}{k} \: = 24k + 8 \\ \\ or \: \: 24 {k}^{2} + 8k - 16 = 0 \\ \\ or \: \: k \: = \frac{ - 8 + - \sqrt{ {8}^{2} - 4 \times 24 \times ( - 16 } }{2 \times 24} \\ \\ or \: \: k = \frac{ - 8 + - \sqrt{64 + 1536} }{48} \\ \\ or \: \: k \: = \frac{ - 8 + - \sqrt{1600} }{48} \\ \\ so \: \: k = \frac{ - 8 + - 40}{48} \\ \\ so \: \: k \: = - 1 \: and \: \frac{32}{48} = \frac{2}{3}$
Hope it helps.

Thanks.

Answer 2

From the above picture you will get the answer