Find sum of roots squared

Question

If $α$, $β$, $γ$, are the roots of the equation $x^3-5x+3=0$, find the value of $α^2+β^2+γ^2$.

Im stuck on this question for a while now, please help me with explanations, thank you very much!

Answer 1

Use Vieta's formula:

Note that: $$(\alpha^2+\beta^2+\gamma^2)=(\alpha+\beta+\gamma)^2-2(\alpha \beta + \alpha \gamma + \beta \gamma)$$

Vieta's formula suggests for a polynomial $ax^3+bx^2+cx+d=0$:

$$(\alpha+\beta+\gamma)=-\frac{b}{a}$$

And:

$$(\alpha \beta + \alpha \gamma + \beta \gamma)=\frac{c}{a}$$

Can you take it from here?

Answer 2

Hint:

$$(\alpha^2+\beta^2+\gamma^2)=(\alpha+\beta+\gamma)^2-2(\alpha\beta+\beta\gamma+\alpha\gamma)$$

Edit:

Hint 2:

Try to expand

$$(x-\alpha)(x-\beta)(x-\gamma)$$