# Solve a cubic equation? [duplicate]

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!

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

• 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. – Tumbleweed May 15 '13 at 6:58
• No derivatives please. – missiledragon May 15 '13 at 7:01

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.

• I can't use derivatives. – missiledragon May 15 '13 at 7:00
• Then use $P(x) = 5(x-a)^2(x-b)$ and solve for $a$ and $b$, by comparing coefficients. – AnalyseThat May 15 '13 at 7:01

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.

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 $a\cdot a\cdot b=-\frac{54}5$

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