Question

If the sum of the zeros of the polynomial x^2-(k-4)X+2(4k-7) is half of their product, then the value of k is

Answer 1

$\bold \red \star \: \large \underline{Solution:-}$

We have,

$\\ {x}^{2} - (k - 4)x + 2(4k + 7).$

$\bold{Sum \: of \: Roots} \\ \\ \: \: \: \: \: \: \: \red \star \: \: \alpha + \beta = \frac{ - b}{a} . \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = \frac{ \cancel{-} \{\cancel{-} (k - 4) \} )}{1} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = k - 4.$

$\bold{Product \: of \: Roots} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \red\star \: \: \alpha \times \beta = \frac{c}{a} . \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = 2(4k - 7).$

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$\\ \\ \tt \bold{According \: to \: condition}$

$\alpha + \beta = \frac{1}{2} \times \alpha \times \beta . \\ \\ k - 4 = \frac{1}{ \cancel2} \times \cancel{2}(4k - 7) \\ \\ k - 4 = 4k - 7. \\ \\ - 4 + 7 = 3k. \\ \\ 3 = 3k. \\ \\ k = \frac{ \cancel3}{ \cancel3} . \\ \\ k = 1.$

Hence,

$\tt \huge{ The\: value \:of\: k = \red{1}.}$