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

AI Thread Summary
To find the value of x1^6 + x2^6 for the quadratic equation 25x^2 - 5√76x + 15 = 0 without solving it, one can use the relationships derived from Vieta's formulas. The sum of the roots x1 + x2 is calculated as -b/a, and the product x1 * x2 as c/a. The discussion highlights that factoring the quadratic does not equate to solving it, and completing the square is suggested as a method. The participants emphasize the importance of deriving the necessary values from the coefficients rather than finding the roots directly. This approach allows for the calculation of x1^6 + x2^6 using the established relationships.
chloe1995
Messages
2
Reaction score
0

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.
 
Physics news on Phys.org
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.
 
chloe1995 said:

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}\ .\
 
SteamKing said:
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.

SammyS said:
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.
 
So, Have you managed to solve the problem?
 

Thread 'Family of lines that are at a distance of 5 from the origin'

I picked up this problem from the Schaum's series book titled "College Mathematics" by Ayres/Schmidt. It is a solved problem in the book. But what surprised me was that the solution to this problem was given in one line without any explanation. I could, therefore, not understand how the given one-line solution was reached. The one-line solution in the book says: The equation is ##x \cos{\omega} +y \sin{\omega} - 5 = 0##, ##\omega## being the parameter. From my side, the only thing I could...
View full post »

Thread 'Making the denominator of a certain fraction real'

Essentially I just have this problem that I'm stuck on, on a sheet about complex numbers: Show that, for ##|r|<1,## $$1+r\cos(x)+r^2\cos(2x)+r^3\cos(3x)...=\frac{1-r\cos(x)}{1-2r\cos(x)+r^2}$$ My first thought was to express it as a geometric series, where the real part of the sum of the series would be the series you see above: $$1+re^{ix}+r^2e^{2ix}+r^3e^{3ix}...$$ The sum of this series is just: $$\frac{(re^{ix})^n-1}{re^{ix} - 1}$$ I'm having some trouble trying to figure out what to...
View full post »

Similar threads

Cubic equation solutions form
Replies
2
Views
2K
Both roots of a quadratic equation lying within limits
Replies
22
Views
3K
Three () distinct real roots of a quadratic equation
Replies
27
Views
4K
Find the value of ##r## and ##s## in the given quadratic equation
Replies
1
Views
2K
Find integer points on this equation
Replies
8
Views
1K
Solving system of 3 quadratic equations
Replies
4
Views
2K
A gardener collected 17 apples...
Replies
32
Views
2K
Value of "a" between two real roots of quadratic equation
Replies
2
Views
2K
Finding the solution to system of 3 equations with 3 unknowns
Replies
2
Views
1K
Integer programming model (alternating constraints)
Replies
5
Views
2K

Hot Threads

Recent Insights

Back
Top