Expression involving roots of quadratic equation

[mathdad]

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.

 

[Greg]

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. :)

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 $-\frac{b}{a}$ and the product of the roots is $\frac{c}{a}$.

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

That expression evaluates to $\frac{b^2-2ac}{c^2}$.