Solution to 4. If the roots of the cubic az3 + bz2 + cz + d = … - Sikademy
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4. If the roots of the cubic az3 + bz2 + cz + d = 0 form an arithmetic progression α − β, α, α + β, prove that (2b 2 − 9ac)b + 27a 2d = 0.

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By the Vieta's formulas we have -\frac{b}{a}=(\alpha-\beta)+\alpha+(\alpha+\beta)=3\alpha\frac{c}{a}=(\alpha-\beta)\alpha+(\alpha-\beta)(\alpha+\beta)+\alpha(\alpha+\beta)=

=\alpha^2-\alpha\beta+\alpha^2-\beta^2+\alpha^2+\alpha\beta=3\alpha^2-\beta^2-\frac{d}{a}=(\alpha-\beta)\alpha(\alpha+\beta)=\alpha^3-\alpha\beta^2

Then (2b^2-9ac)b+27a^2d=a^3\left(2\left(\frac{b}{a}\right)^2-9\frac{c}{a}\right)\frac{b}{a}+27a^3\frac{d}{a}

So we need to prove that \left(2\left(\frac{b}{a}\right)^2-9\frac{c}{a}\right)\frac{b}{a}+27\frac{d}{a}=0

We have 2\left(\frac{b}{a}\right)^2-9\frac{c}{a}=2(3\alpha)^2-9(3\alpha^2-\beta^2)=-9\alpha^2+9\beta^2, so \left(2\left(\frac{b}{a}\right)^2-9\frac{c}{a}\right)\frac{b}{a}+27\frac{d}{a}=

=(-9\alpha^2+9\beta^2)(-3\alpha)+27(-\alpha^3+\alpha\beta^2)=0

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Question ID: mtid-5-stid-8-sqid-3927-qpid-2626