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 'Finding the number of ways to arrange identical balls in a circle (3 different colors)'

I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
View full post »

Thread 'Greatest possible value of a constant in polynomial'

Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks
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