Quadratic Equation to prove $ax^2+bx+c=0$

Question

"Prove that there is one and only quadratic equation for which the sum of the roots is $3$ and the cubed of the roots is $63$"

I'm practicing for the Maths Olympiad. I'm a high school student and it's too hard for me. Can you please solve it for me?

Answer 1

If $a,b$ are the roots,

$$p+q=3$$

$$63=p^3+q^3=(p+q)^3-3pb(p+q)\implies pq=?$$

Answer 2

$$\alpha + \beta = 3 = -\frac{b}{a}$$ $$\alpha^3 + \beta^3 = 63 \Rightarrow (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta) = 63 \Rightarrow \alpha\beta = -4 = \frac{c}{a}$$ $$b=-3a$$ and $$c=-4a$$ So the required unique equation is as follows: $$x^2-3x-4=0$$