Find coefficients of polynomials $f(x)=x^2+ax+b$ and $g(x)=x^2+cx+d$ $(a,b,c,d \in \mathbb{R})$ Roots of polynomial $f(x)=x^2+ax+b$ are cubes of the roots of polynomial $g(x)=x^2+cx+d$. Sum and product of roots of polynomial $g(x)$ are equal. Find coefficients $a,b,c,d$ so that polynomial $f(x)$ has double root, and $g(x)$ doesn't have double root.
From the condition that $f(x)$ has double root, discriminant of quadratic equation $x^2+ax+b=0$ must be zero, $$D=a^2-4b=0,b=\frac{a^2}{4}$$ 
Sum and product of roots of polynomial $g(x)$ are equal, $$x_1^3+x_2^3=x_1^3x_2^3$$
Vieta's formulas for two polynomials,
$$x_1+x_2=-a$$
$$x_1x_2=b$$
$$x_1^3+x_2^3=-c$$
$$x_1^3x_2^3=d$$
If we cube first equation, $$(x_1^3+x_2^3)+3x_1x_2(x_1+x_2)=-a^3$$
$$-c-3ba=-a^3$$
From $b=\frac{a^2}{4}$ $$a^3-4c=0,c=\frac{a^3}{4}$$
From $x_1^3+x_2^3=x_1^3x_2^3$ implies $d=-c$
If we cube second equation, $$x_1^3x_2^3=b^3=\frac{a^6}{64}=d=-c$$
I don't know how to find these coefficients.
Thanks for replies.

Let the two roots of $g(x)$ be $p$ and $q$. Since roots of $f(x)$ are cubes of the roots of $g(x)$,
$$\begin{align*}
p+q &= -c\tag1\\
pq &= d\tag2\\
p^3 + q^3 &= -a\tag3\\
p^3q^3 &= b\tag4
\end{align*}$$
From $(3)$, 
$$\begin{align*}
-a &= p^3 + q^3\\
&= (p+q)(p^2 - pq+q^2)\\
&= (p+q)(p^2 + 2pq + q^2 - 3pq)\\
&= -c[(-c)^2 -3d]\\
a &= c^3 -3cd\tag5
\end{align*}$$
And from $(2)$ and $(4)$, $$b = p^3 q^3 = d^3\tag6$$
Since the sum and the product of roots of $g(x)$ are equal, $$d = -c \tag7$$
Writing $a$ and $b$ in terms of $c$,


*

*Substitute $(7)$ into $(5)$, $$a = c^3 + 3c^2\tag8$$

*Substitute $(7)$ into $(6)$, $$b = -c^3\tag{9}$$
Since $f(x)$ has double roots, by using $(8)$ and $(9)$,
$$\begin{align*}
\Delta_f = a^2 - 4b &= 0\\
(c^3 + 3c^2)^2 + 4c^3 &= 0\\
c^4 (c+3)^2 + 4c^3 &= 0\\
c^3 (c^3 + 6c^2 + 9c + 4) &= 0\\
c^3 (c+1)^2(c+ 4) &= 0\\
\end{align*}$$
If $c \in \{0, -4\}$ and using $(7)$, 
$$\begin{align*}
\Delta_g &= c^2 - 4d\\
&= c^2 + 4c\\
&= c(c+4)\\
&= 0
\end{align*}$$ which means $g(x)$ has a double root, contradicting to the given conditions.
Hence $c = -1$. Then $a = 2$, $b=1$ and $d = 1$.
