Question

if alpha and beta are the roots of x ²+5x-1=0 then find the value of alpha ³ + beta ³​

Answer 1

${x}^{2} + 5x - 1 = 0$

$\alpha + \beta = - \frac{b}{a} = \frac{ - 5}{1} = - 5$

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

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

$( - 5) {}^{3} = { \alpha }^{3} + { \beta }^{3} + 3( - 1)( - 5)$

$- 125 = { \alpha }^{3} + { \beta }^{3} + 15$

${ \alpha }^{3} + { \beta }^{3} = - 125 - 15$

${ \alpha }^{3} + { \beta }^{3} = - 140$

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