<s>[INST] Find the value of $x_1^6 +x_2^6$ of this quadratic equation without solving it I got this question for homework and I've never seen anything similar to it.
Solve for $x_1^6+x_2^6$ for the following quadratic equation where $x_1$ and $x_2$ are the two real roots and $x_1 > x_2$, without solving the equation.
$25x^2-5\sqrt{76}x+15=0$
I tried factoring it and I got $(-5x+\sqrt{19})^2-4=0$
What can I do afterwards that does not constitute as solving the equation? Thanks.
 [/INST] There are good methods described here and I want to add one that comes from complex analysis.
Let be $p(x)=3x^3-8x^2+2x-4$ a polinomial and $S_n=r_1^n+r_2^n+r_3^n$ the sum of its $3$ roots to the power $n$.
We can determine the values of $S_n$ by:


*

*recursion as Mark Bennet answer (very efficient method)

*iteration following a simple division.


Recursion: 
If we change $x^m$ by $S_m$ in the polinomial, it will be equal to zero.
it means: 
       $$ 3S_3-8S_2+2S_1-4S_0=0 $$ 
but it works for all real numbers and all polynomials. 
If $\ell\in\mathbb{R}$, then $ 3S_\ell-8S_{\ell-1}+2S_{\ell-2}-4S_{\ell-3}=0 $. As all knows, $S_0=$ the number of roots (polynomial's degree), $S_1=$ sum of roots, etc.
Iteration:
For this method we need to use the derivative of $p(x)$ and the Horner schema for division:
$$ \dfrac{p'(x)}{p(x)}=S_0+S_1x+S_2x^2+S_3x^4+\dots $$
In this case, we see that $p'(x)=9x^2-16x+2$. Then:
Division by Horner of p'(x)/p(x) 

              3| 9    -16    2         ---> p'(x)
  p(x)-->     8|       24   -6    12 
             -2|       
              4|____________________________
               | S_0  S_1   S_2   S_3   S_4  ...   

Why those methods works?
Recursive method words because $3S_3-8S_2+2S_1-4S_0= 3(r_1^3+r_2^3+r_3^3) -8(r_1^2+r_2^2+r_3^2)+2(r_1+r_2+r_3)-4(r_1^0+r_2^0+r_3^0) $
and this is obviously equal to: 
$p(r_1)+p(r_2)+p(r_3)=0+0+0=0.  $ The same if you multiply all by $r^{\ell-3}$. You will have the same result.
Iterative methods works because complex calculus is overpowerfull.
$p(x)=3(x-r_1)(x-r_2)(x-r_3)$ so $\ln(p(x))= \ln(3)+\ln(x-r_1)+\ln(x-r_2)+\ln(x-r_3)  $
a simple derivative gives us:
$$\dfrac{p'(x)}{p(x)} = 0 + \dfrac{\color{red} 1}{x-r_1} + \dfrac{\color{red} 1}{x-r_2}+ \dfrac{\color{red} 1}{x-r_3} = \displaystyle\sum_{1\le i\le 3} \left( \dfrac{1}{1-\frac{r_i}{x}} \right) $$
(the red one's will change for the multiplicity of the root)
and $$  \displaystyle\sum_{1\le i\le 3} \left( \dfrac{1}{1-\frac{r_i}{x}} \right) =  \displaystyle\sum_{1\le i\le 3} \left( 1+\frac{r_i}{x}+\frac{r_i^2}{x^2}+\frac{r_i^3}{x^3}+\cdots \right) = (1+1+1) + \frac{r_1+r_2+r_3}{x} + \frac{r_1^2+r_2^2+r_3^2}{x^2}  + \cdots $$
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