Question

If aplha and beta are the roots of x²=x+1 then value of alpha²/beta - beta²/alpha is ​

Answer 1

Answer:

-2√5

Step-by-step explanation:

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Answer 2

p(x) =

${x}^{2} - x - 1 = 0$

Here, a=1 ,b=-1and c=-1

As we know that,

$\alpha + \beta = \frac{ - b}{a } = \frac{ - ( -1 )}{1} = 1$

and

$\alpha \beta = \frac{c}{a} = \frac{ - 1}{1} = - 1$

To find

$\frac{ { \alpha }^{2} }{ \beta } - \frac{ { \beta }^{2} }{ \alpha } \\ = > \frac{ { \alpha }^{3} - { \beta }^{3} }{ \alpha \beta } \\ = > \frac{( \alpha - \beta )( { \alpha }^{2} + { \beta }^{2} + \alpha \beta )}{ \alpha \beta }$

$= > \frac{ \sqrt{( \alpha + \beta ) ^{2} - 4 \alpha \beta } ( ( \alpha + \beta ) ^{2} - 2 \alpha \beta + \alpha \beta )}{ \alpha \beta } \\ = > \frac{ \sqrt{ {1}^{2} - 4( - 1) } (1^{2} - 2( -1) + ( - 1) }{ - 1} \\ = > \frac{ \sqrt{5}(2) }{ - 1} \\ = > - 2 \sqrt{5}$

Hence the value is

$\boxed{ \red{ - 2 \sqrt{5} }}$