# solving an equation

• Aug 30th 2007, 06:43 PM
Mr_Green
solving an equation
Determine k and solve the equation x^3 -kx^2 + 20x +12 = 0.

Thanks
• Aug 30th 2007, 06:45 PM
ThePerfectHacker
Quote:

Originally Posted by Mr_Green
Determine k and solve the equation x^3 -kx^2 + 20x +12 = 0.

Thanks

What is the condition on this equation.
• Aug 30th 2007, 06:47 PM
Mr_Green

Determine k and solve the equation x^3 - kx^2 +20x + 12 = 0, if one of its zeros is the triple of another.
• Aug 30th 2007, 07:03 PM
ThePerfectHacker
Quote:

Originally Posted by Mr_Green

Determine k and solve the equation x^3 - kx^2 +20x + 12 = 0, if one of its zeros is the triple of another.

Let $\displaystyle a,b,c$ be zero's (counting multiplicity).
One of its zeros, say $\displaystyle c$, is triple of another, say $\displaystyle a$. Thus, $\displaystyle c=3a$.

Thus, we have that,
$\displaystyle \left\{ \begin{array}{c}a+b+3a = k\\ab+3a^2+3ab=20\\ 3a^2b= -12 \end{array}\right.$

Thus,
$\displaystyle \left\{ \begin{array}{c}4a+b = k\\ 4ab+3a^2 = 20 \\ a^2b = -4 \end{array}\right.$

Now we solve for $\displaystyle a,b,k$ ...

But, I have an easier way. Solving for that equation can get messy. Instead, let us hope that its solutions are integers. Not necessarily this approach might work but let us try it, if it works then we can save a lot of work. If it does not we can try another way.

Note,
$\displaystyle a^2b=-4$.
Now, $\displaystyle a^2>0$ so $\displaystyle b<0$. It must be that $\displaystyle a^2 \mbox{ and }b$. Are factors of $\displaystyle 4$. Since $\displaystyle a^2$ must be a square and positive. It means $\displaystyle a^2 = 4 \mbox{ or }1$. Thus, the solutions are $\displaystyle (a^2,b) = (4,-1) \mbox{ or } (1,-4)$. Now take the square root to remove the square on $\displaystyle a$ to get: $\displaystyle (a,b) = (-2,-1), (2,-1), (-1,-4) , (1,-4)$.
Now which of these pairs satisfies equation 2, i.e. $\displaystyle 4ab+3a^2 = 20$.
We easily see that $\displaystyle (-2,-1)$ is the only solution.
Thus, $\displaystyle a=-2,b=-1,c=3(a)=-6$.
Thus, $\displaystyle k = -(a+b+c) = 9$.