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

  • Thread starter chloe1995
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Homework Statement



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]

Homework Equations


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.
 

Answers and Replies

  • #2
SteamKing
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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.
 
  • #3
SammyS
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Homework Statement



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]

Homework Equations


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.
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]
 
  • #4
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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, [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]
Thank you.
 
  • #5
SammyS
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So, Have you managed to solve the problem?
 

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