Question

If alpha and Beta are the zeros of the quadratic polynomial p(x) = 4x2 - 5x - 1, find the value of alpha2 beta+beta2 alpha??
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Answer 1

Given:

$\star \bold{ \alpha \: \: and \: \: \beta \: are \: \: the \: zeroes \: of \: the} \\ \bold{quadratic \: polynomial \: \:} \\ \bold{ p(x) = 4 {x}^{2} - 5x - 1 }$

To find :

$\star \: { \alpha }^{2} \beta + { \beta }^{2} \alpha$

Solution :

• p ( x ) = 4x² - 5x - 1

Compare with ax² + bx +c , we get

• a = 4 , b = -5 and c = -1 .

$\leadsto \: \alpha + \beta = \frac{ - b}{a} = \frac{ - ( - 5)}{4} = \frac{5}{4} \\ \\ \leadsto \: \alpha \beta = \frac{c}{a} = \frac{ - 1}{4}$

Now ,

$\leadsto \: { \alpha }^{2} \beta + { \beta }^{2} \alpha \\ \\ \leadsto \: \alpha \beta ( \alpha + \beta ) \\ \\ \leadsto \: \frac{ - 1}{4} ( \frac{5}{4} ) \\ \\ \leadsto \: \frac{ - 5}{16}$

Hence , the value is -5/16.