Question

If alpha,beta are the roots of x2 – px +q=0,
then alpha^3+beta^3=
​

Answer 1

Answer:

$\alpha^3 + \beta^3$  = $p^3$- 2$q^2$

Step-by-step explanation:

Given equation  $x^2$- px+q = 0

so, a= 1 ; b = -p; c= q

Now α+β=  -b/a  = p

     αβ = c/a = q

now (α+β$)^2$ = (p$)^2$

or $\alpha^2 + \beta^2 + 2\alpha\beta$ = $p^2$

or $\alpha^2 + \beta^2$ =  $p^2$ - 2q

also $\alpha^3 + \beta^3$ = ($\alpha+ \beta$)($\alpha^2 + \beta^2 - \alpha\beta$)

$\alpha^3 + \beta^3$ = (p)( $p^2$-2q -q)

Therefore  $\alpha^3 + \beta^3$  = $p^3$- 2$q^2$

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