Question

$if \alpha and \beta are \: two \: polynomials \: of \: f{x} = kx^{2} + 4x + 4 \: satisfyig \: the \: relation \: \ { \alpha }^{2} + { \beta }^{2} = 24findthe \: value \: of \: k$
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Answer 1

QUESTION

If α and β are two zeroes of polynomial f(x) = kx² + 4x + 4 satisfying the relation α² + β² = 24 . find the value of k.

SOLUTION

∵ α and β are zeroes of polynomial kx² + 4x + 4

∴ α + β = - b / a =   - 4 / k

and

αβ = c / a = 4 / k

now,

∵ ( x + y )² = x² + y² + 2 x y

so, x² + y² = ( x + y )² - 2 x y

∴  α² + β² =  ( α + β )² - 2 α β

$24 = (\frac{-4}{k} )^2 - 2 ( \frac{4}{k} )\\\\24 = \frac{16}{k^2} - \frac{8}{k} \\\\( taking \ LCM)\\\\ 24 = \frac{16 - 8k}{k^2} \\\\( cross \ multiplying )\\\\24 k^2 = 16 - 8 k\\\\24 k ^2 + 8 k - 16 = 0\\\\( dividing \ by \ 8\ both\ sides ) \\\\3k^2 + k - 2 = 0\\\\3k^2 + 3 k - 2 k -2= 0\\\\3k(k+1) -2(k+1)=0\\\\(3k-2)(k+1)=0\\\\k = 2 / 3 \\and \\k = -1$

HENCE THERE ARE TWO POSSIBLE VALUES OF k

k = 2 / 3 and  k = -1 .