Question

if alpha and beta are the zeros of quadratic polynomial 4 x square - 5 x minus 1 find the value of alpha/beta + beta/ alpha​

Answer 1

Step-by-step explanation:

$\bold{Correct \: Question -} \\ If \: \alpha \: and \: \beta \: are \: the \: Zeroes \: of \: \\ quadratic \: polynomial \: 4{x}^{2} - 5x + 1 \\ then, \: Find \: the \: value \: of \: \frac{ \alpha}{ \beta} + \frac{ \beta}{ \alpha}$

$\bold{Given-} \\ \bold{\alpha }\: and \: \bold{\beta} \: are \: zeroes \: of \: quadratic \: polynomial \\ \bold{4 {x}^{2} - 5x + 1} \\ \\ \bold{To \: Find - } \\ \bold{ Value \: of \: \: \frac{ \alpha}{ \beta} + \frac{ \beta}{ \alpha} } \\ \\ Now, \\ According \: to \: the \: Question \\ \\ At \: first \: we \: need \: to \: \bold{Factorise} \\ \bold{4 {x}^{2} - 5x + 1} \: to \: find \: zeroes \: of \: this \: polynomial \\ \\ 4 {x}^{2} - 5x + 1 \\ 4 {x}^{2} - 4x - x + 1 \\ 4x(x - 1) - 1(x - 1)\\(4x - 1)(x - 1) \\ \\ \bold{Zeroes \: are - } \\ 4x - 1 = 0 \: \: \: and \: \: x - 1 = 0 \\ \fbox \bold{x = \frac{1}{4} \: \: \: and \: \: x = 1}$

$Let \: \bold{\alpha = 1} \\ And \\ \bold{ \beta = \frac{1}{4} } \\ \\ Now, \\ We \: need \: to \: find \: the \: value \: of \\ \bold{ \frac{ \alpha}{ \beta} + \frac{ \beta}{ \alpha} } \\ \\ \frac{1}{ \frac{1}{4} } + \frac{ \frac{1}{4} }{1} \\ 4 + \frac{1}{4} \\ \frac{16 + 1}{4} \\ \\ = \frac{17}{4} \\ \\ Hence, \\ The \: value \: of \: \bold{ \frac{ \alpha}{ \beta} + \frac{ \beta}{ \alpha} } \: is \: \bold{ \frac{17}{4} }$

Answer 2

$\huge \tt \it \bf \it \bm { \mathbb{ \fcolorbox{blue}{yellow}{ \red{ANSWER :}}}} \green\longrightarrow$

$\huge \green { \underline{GIVEN}} \green\longrightarrow$

$\large4 {x}^{2} - 5x - 1$

$\huge \green { \underline{TO \ FIND}} \green\longrightarrow$

$\large\frac{ \alpha }{ \beta } + \frac{ \beta }{ \alpha }$

$\huge \green{ \underline{SOLUTION}} \green\longrightarrow$

$\longrightarrow4 {x}^{2} - 5x + 1 \\ \\ \longrightarrow4 {x}^{2} - 4x - x + 1 \\ \\\longrightarrow 4x(x - 1) - 1(x - 1) \\ \\ \longrightarrow (4x - 1)(x - 1)$

$\huge \green{ \underline{ ZEROES \: ARE}} \longrightarrow$

$\large \boxed{ \fbox{ \red{x = \frac{1}{4} \: and \: x = 1 }}}$

NOW,

$\large \frac{ \alpha }{ \beta } + \frac{ \beta }{ \alpha } = \frac{ \frac{1}{1} }{4} + \frac{ \frac{1}{4} }{1} \\ \\ = > \frac{4}{1} + \frac{1}{4} \\ \\ = > \frac{16 + 1}{4} \\ \\ = > \frac{17}{4}$

so,

$\large{ \boxed{ \fbox{ \blue{ \frac{ \alpha }{ \beta } + \frac{ \beta }{ \alpha } = \frac{17}{4} }}}}$

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