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

[chloe1995]

Homework Statement

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.

$25x^2-5\sqrt{76}x+15=0$

Homework Equations

The Attempt at a Solution

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.

 

[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]

Homework Statement

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.

$25x^2-5\sqrt{76}x+15=0$

Homework Equations

The Attempt at a Solution

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.

Hello chloe1995. Welcome to PF !

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

Then $\displaystyle \ \ x_1 + x_2 = -\frac{b}{a}\ \$ and $\displaystyle \ \ x_1\cdot x_2=\frac{c}{a}\ .\$

 

[chloe1995]

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.

Oops! I meant completing the square.

Hello chloe1995. Welcome to PF !

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

Then $\displaystyle \ \ x_1 + x_2 = -\frac{b}{a}\ \$ and $\displaystyle \ \ x_1\cdot x_2=\frac{c}{a}\ .\$

Thank you.

 

[SammyS]

So, Have you managed to solve the problem?