Solving Cubic Equations with Lagrange Resolvent?

I'm having difficulties understanding my textbook's decription of solving cubic equations using Lagrange Resolvents and symmetric polynomials.

Here's what I understand:

$${x}^{3}+px-q=(x-r)(x-s)(x-t)$$

We can also write:

$$\lambda =r+ws+{w}^{2}t$$

$$\mu =wr+s+{w}^{2}t$$

where $1,w,{w}^{2}$ are the cubic roots of 1. I then understand that ${\lambda}^{2}+{\mu}^{3}$ and ${\lambda}^{3}{\mu}^{3}$ are symmetric polynomials in r, s, and t. It is also solvable that the elemntary symmetric functions in r, s, t are 0,p,q where

$$r+s+t=0$$

$$rs+rt+st=p$$

$$rst=q$$

The part where I get confused is that the book claims that ${\lambda}^{3}$ and ${\mu}^{3}$ are the roots of the quadratic polynomial $q(x)={x}^{2}-({\lambda}^{3}+{\mu}^{3})x+{\lambda}^{3}{\mu}^{3}$, which seems obvious to me. Then they claim you can use the quadratic formula to solve for ${\lambda}^{3}$ and ${\mu}^{3}$ in terms of p and q, thus allowing you to solve a system of equations to acquire r,s,t.

How can you use the quadratic formula to "explicitly solve for ${\lambda}^{3}$ and ${\mu}^{3}$ in terms of p and q"?

I'm having difficulties understanding my textbook's decription of solving cubic equations using Lagrange Resolvents and symmetric polynomials.

Here's what I understand:

$${x}^{3}+px-q=(x-r)(x-s)(x-t)$$

We can also write:

$$\lambda =r+ws+{w}^{2}t$$

$$\mu =wr+s+{w}^{2}t$$

where $1,w,{w}^{2}$ are the cubic roots of 1. I then understand that ${\lambda}^{2}+{\mu}^{3}$ and ${\lambda}^{3}{\mu}^{3}$ are symmetric polynomials in r, s, and t. It is also solvable that the elemntary symmetric functions in r, s, t are 0,p,q where

$$r+s+t=0$$

$$rs+rt+st=p$$

$$rst=q$$

The part where I get confused is that the book claims that ${\lambda}^{3}$ and ${\mu}^{3}$ are the roots of the quadratic polynomial $q(x)={x}^{2}-({\lambda}^{3}+{\mu}^{3})x+{\lambda}^{3}{\mu}^{3}$, which seems obvious to me. Then they claim you can use the quadratic formula to solve for ${\lambda}^{3}$ and ${\mu}^{3}$ in terms of p and q, thus allowing you to solve a system of equations to acquire r,s,t.

How can you use the quadratic formula to "explicitly solve for ${\lambda}^{3}$ and ${\mu}^{3}$ in terms of p and q"?