Question

if alpha and beta are zeroes of polynomial f(x)=x^2-x+1=0 then find the value of alpha^2+beta^2. answer fast please.....

Answer 1

`{ \boxed{ \mathfrak{amswer}}} :{ \underline{ \tt{ \: { \alpha }^(2) + { \beta }^(2) = - 1} \: }}`

Explanation:

From General equation of a polynomial:

✍️ x² - (Sum of roots)x + (Product of roots) = 0

Therefore;

`{ \tt{f(x) = {x}^(2) - x + 1 = 0 }}`

From the equation,

• Sum = 1
• Product = 1

`{ \tt{ \alpha + \beta = 1}} - - - { \rm{sum}} \\ { \tt{ \alpha \beta = 1 }} - - - { \rm{product}}`

From the question:

`{ \tt{ {( \alpha + \beta )}^(2) = ( \alpha + \beta )( \alpha + \beta ) }} \\ \\ { \tt{ {( \alpha + \beta )}^(2) = { \alpha }^(2) + 2 \alpha \beta + { \beta }^(2) }} \\ \\ { \tt{ {( \alpha + \beta )}^(2) = ( { \alpha }^(2) + { \beta }^(2) ) + 2 \alpha \beta }} \\ \\ { \boxed{ \tt{( { \alpha }^(2) + { \beta }^(2) ) = {{( \alpha + \beta )}^(2) - 2 \alpha \beta }}}}`

Therefore, from sum and product

`{ \tt{ { \alpha }^(2) + { \beta }^(2) = (1) {}^(2) - 2(1) }} \\ = { \tt{1 - 2}} \\ = - 1`