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If $\alpha$ and $\beta$ are the roots of the equation $$x^2 + x − 3 = 0$$ find the value of the expression $4\beta^2 − \alpha^3$.

I tried using sum of roots and product of roots formulas but could not get the answer.

amWhy
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KBC
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    Well, the quadratic formula gives you closed forms for the roots, so you could just compute it. – lulu Jul 18 '17 at 17:39
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    I honestly don't see how you could get stuck (mainly since you haven't shown us your work) What's holding you back from computing the roots directly, for example? – Simply Beautiful Art Jul 18 '17 at 17:41
  • Please add your attempts, even if you got stuck in them, for each approach you state you tried using. – amWhy Jul 18 '17 at 17:59
  • Your question is incomplete; please add the attempts you've claimed (actual workings, not claims). And Hmmm, have you tried using the quadratic equation? – amWhy Jul 18 '17 at 18:11

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We have $$\beta^2=3-\beta\\4\beta^2=12-4\beta$$ Also $$\alpha^2=3-\alpha\\\alpha^3=3\alpha-\alpha^2=3\alpha-3+\alpha=4\alpha-3$$ Thus, $$4\beta^2-\alpha^3=15-4(\alpha+\beta)=15+4=19$$

Sahiba Arora
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