MHB Expression involving roots of quadratic equation

AI Thread Summary
To evaluate the expression $\frac{1}{\alpha^2}+\frac{1}{\beta^2}$ given the roots $\alpha$ and $\beta$ of a quadratic equation, the discussion highlights that this can be rewritten as $\frac{\alpha^2+\beta^2}{\alpha^2\beta^2}$. The sum of the roots is expressed as $-\frac{b}{a}$ and the product as $\frac{c}{a}$. The derived expression ultimately simplifies to $\frac{b^2 - 2ac}{c^2}$. This approach effectively connects the roots of the quadratic equation to the desired expression.
mathdad
Messages
1,280
Reaction score
0
Given that $\alpha$ and $\beta$ are the roots of a quadratic equation, evaluate $\frac{1}{\alpha^2}+\frac{1}{\beta^2}$.

View attachment 7497

I find this question to be interesting.
 

Attachments

  • MathMagic171110_2.png
    MathMagic171110_2.png
    14.6 KB · Views: 103
Last edited by a moderator:
Mathematics news on Phys.org
I've edited your post to include the problem statement in the body of the post and to give it a meaningful title. Please try to do so on your own in the future. :)

RTCNTC said:
Given that $\alpha$ and $\beta$ are the roots of a quadratic equation, evaluate $\frac{1}{\alpha^2}+\frac{1}{\beta^2}$.
$$\gamma(x-\alpha)(x-\beta)=\gamma x^2-\gamma(\alpha+\beta)x+\gamma\alpha\beta$$

Using $ax^2+bx+c$, the sum of the roots is $-\frac{b}{a}$[/size] and the product of the roots is $\frac{c}{a}$[/size].

The given expression with a common denominator is $\frac{\alpha^2+\beta^2}{\alpha^2\beta^2}$[/size].

That expression evaluates to $\frac{b^2-2ac}{c^2}$[/size].
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top