Question

Formula relațiilor lui Viete x1^3+x2^3.
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Answer 1

$ax^2+bx+c = 0\Big|\cdot (x\neq 0) \\ \\ \Leftrightarrow\,ax^3+bx^2+cx\\ \Leftrightarrow\,ax^3 = -bx^2-cx\\ \Leftrightarrow\, x^3= -\dfrac{b}{a}x^2 - \dfrac{c}{a}x\\ \\ x_1^2+x_2^2 = (x_1+x_2)^2 - 2x_1x_2 = \Big(-\dfrac{b}{a}\Big)^2-2\dfrac{c}{a} = \dfrac{b^2-2ac}{a^2}$

$\Rightarrow x_1^3+x_2^3 = -\dfrac{b}{a}(x_1^2+x_2^2)-\dfrac{c}{a}(x_1+x_2)\\\\\Rightarrow x_1^3+x_2^3 = -\dfrac{b}{a}\left(\dfrac{b^2-2ac}{a^2}\right)-\dfrac{c}{a}\cdot \left(-\dfrac{b}{a}\right)\\ \\ \Rightarrow x_1^3+x_2^3 =\dfrac{-b^3+2abc}{a^3}+\dfrac{bc}{a^2}\\ \\ \Rightarrow \boxed{x_1^3+x_2^3 = -\dfrac{b^3-3abc}{a^3}}$