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This question already has an answer here:

Need help with solving an equation:

Solve the equation $5x^3 - 24x^2 + 9x + 54 = 0$ given that two of it's roots are equal.

Any help would be greatly appreciated. Thanks!

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marked as duplicate by mercio, Peter Taylor, 23rd, Lord_Farin, Stahl May 15 '13 at 7:37

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ wolframalpha.com/input/?i=5x3%E2%88%9224x2%2B9x%2B54%3D0 This may not what you need, but wolfram alpha it is something good to know. $\endgroup$ – Tumbleweed May 15 '13 at 6:58
  • $\begingroup$ No derivatives please. $\endgroup$ – missiledragon May 15 '13 at 7:01
  • $\begingroup$ I'm downvoting because of the duplicate. People are here to help you learn, not give out free answers. Please show more respect for the community. $\endgroup$ – 6005 May 15 '13 at 7:40
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If two roots are equal, then the derivative will have that as a root.

If you can't use derivatives, you can set $P(x) = 5(x-a)^2(x-b)$ and solve for $a$ and $b$, by comparing coefficients.

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  • $\begingroup$ I can't use derivatives. $\endgroup$ – missiledragon May 15 '13 at 7:00
  • $\begingroup$ Then use $P(x) = 5(x-a)^2(x-b)$ and solve for $a$ and $b$, by comparing coefficients. $\endgroup$ – AnalyseThat May 15 '13 at 7:01
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If you want to use the fact that two roots are equal, then we have that if $a$ is a double root of $f(x)$, then $a$ is a root of $f'(x)$. Use this to narrow down the roots.

You could also use the rational root test to see if there are some trivial roots, we can conclude directly.

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Using Vieta's Formula,

$a\cdot a+a\cdot b+a\cdot b=\frac95\implies 2ab+a^2=\frac95$

$a+a+b=\frac{24}5\implies 2a+b=\frac{24}5\implies 4a^2+2ab=\frac{48}5a$

Comparing the values of $a\cdot b$ and on simplification we get, $5a^2-16a+3=0$

Solve for $a,$ find the corresponding $b$ which satisfies $a\cdot a\cdot b=-\frac{54}5$

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  • $\begingroup$ What is wrong with this method? $\endgroup$ – lab bhattacharjee May 18 '13 at 11:02

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