Question

$f(x) = {x }^{2} - x - 2. \: find \: \alpha {}^{2} + { \beta }^{2} \: \: \: \: \: { \alpha }^{3} + { \beta }^{3} \: \: \: \: \: \alpha \div \beta + \beta \div \alpha$
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Answer 1

Answer :-

➡️ Given Quadratic Equation :-

▪️$\sf{ x^2 - x - 2}$

◾And roots of equation :-

▪️$\sf{\alpha \: and \: \beta}$

◾Now

▪️Sum of roots = $\sf{\dfrac{-b}{a} = 1 = (\alpha + \beta)}$

▪️Product of roots = $\sf{\dfrac{c}{a} = -2 = (\alpha \times \beta) }$

◾Now

→$\sf{\alpha^2 + \beta^2 }$

$\sf{= (\alpha + \beta)^2 - 2\alpha\beta }$

$\sf{= (1)^2 - 2(-2) }$

$\sf{= 1 + 4 }$

$\sf{ = 5}$

→$\sf{\alpha^3 + \beta^3}$

$\sf{= (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta) }$

$\sf{= (1)^3 - 3(-2)(1) }$

$\sf{= 1 + 6 }$

$\sf{ = 7}$