Solving a quadratic using vietas theorem I keep going in circles.

Question

I am trying to solve the quadratic equation $x^2-48x+432=0$ with out directly factoring OR using the quadratic equation. I am going with vieta. So $$r+s=48$$ and $$rs=432$$ I've already solved it by plugging in factors of $432$ the roots are $$r=12, s=36 $$

but that is too much guessing so I wanted to try a new approach.

Using the fact that $s=\frac{432}{r}$ and subbing in the sum gets me. $$r+\frac{432}{r}=48$$

Here lies my problem. Manipulating the equation to solve for $r$ gets me back to the original quadratic. What am I missing? Or is plugging the only way to do this?

Answer 1

Complete the square $$x^2-48x+432=0\iff x^2-2\cdot x\cdot24+24^2=576-432$$

$$\iff(x-24)^2=(12)^2$$

Answer 2

$(r-s)^2 = (r+s)^2 - 4rs$

So you can get $(r-s)$ and hence $r = \frac{(r+s)+(r-s)}{2}$

Answer 3

Given a quadratic equation $x^2-bx+c=0$, then solving $r+s=b$ and $rs=c$ is equivalent to finding $r,s$ such $(x-r)(x-s)=x^2-bx+c$. So, Vieta's formulas are nice but do not simplify anything.