Find the value of x1^6 +x2^6 of this quadratic equation without solving it

chloe1995

1. The problem statement, all variables and given/known data

Solve for [itex]x_1^6+x_2^6[/itex] for the following quadratic equation where [itex]x_1[/itex] and [itex]x_2[/itex] are the two real roots and [itex]x_1 > x_2[/itex], without solving the equation.

[itex]25x^2-5\sqrt{76}x+15=0[/itex]

2. Relevant equations
3. The attempt at a solution

I tried factoring it and I got [itex](-5x+\sqrt{19})^2-4=0[/itex]

What can I do afterwards that does not constitute as solving the equation? Thanks.

SteamKing

Notice that 5 can be factored from the quadratic without changing the roots.

Also, you haven't truly factored the quadratic, you have merely re-written it.

SammyS

Hello chloe1995. Welcome to PF !

Suppose that x₁ and x₂ are the solutions to the quadratic equation, [itex]\displaystyle \ \ ax^2+bx+c=0\ .[/itex]

Then [itex]\displaystyle \ \ x_1 + x_2 = -\frac{b}{a}\ \ [/itex] and [itex]\displaystyle \ \ x_1\cdot x_2=\frac{c}{a}\ .\ [/itex]

chloe1995

Oops! I meant completing the square.

Thank you.

SammyS

So, Have you managed to solve the problem?