Prove the expression by using Quadratic equation's condition

Question

One root of quadratic equation $a*x^2+b*x+c$ is square of another root. Then prove that $c*(a-b)^2 = (b^2-ac)*(a-b)$.

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score 0 · Answer 1 · answered Sep 14 '17 at 06:44

By Vieta's relations,

$$-\frac ba=r+r^2,\\\frac ca=r^3$$ and you have to eliminate $r$.

Then

$$1-\frac ba=1+r+r^2=\frac{r^3-1}{r-1}=\frac{\dfrac ca-1}{r-1}$$ gives you $r=\dfrac{c-b}{a-b}$, which you plug in one of the equations.

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