I have a probably very simple problem here. A system of nonlinear equations.
$$\left\{ \begin{align} & {{x}^{2}}+{{y}^{2}}=26 \\ & x+{{y}^{2}}=6 \\ \end{align} \right.$$
I started with the first equation;
$$x = - y^2 + 6$$
I substituted it into the second equation;
$$(-y^2 + 6)^2 + y^2 = 26$$ $$y^4 - 11y^2 + 36 = 26$$
At which point I am now stuck. Still solving for my 'x' yet, the further I go, it does not seem to make sense. I would use a quadratic formula, would that be suggestible? Given my highest multiplicity is larger than 2? Is it even relevant what my multiplicities are.
Thank you in advance.
P.S. Please excuse my absent use of mathjax typeset. I am no pro and also the reference page I use to write from is down at the moment.
Yes, you can use the quadratic formula if you first substitute $Y=y^2$ into $y^4 - 11y^2 + 36 = 26$. The resulting equation will be $ Y^2-11Y+10=0$. Now solve for $Y$ and note that $y=\pm \sqrt{Y}$.
However, it might be easier to substitute $y^2=-x+6$ into $x^2+y^2=26$ to get $$x^2-x-20=0$$ Now you can solve for $x$ using the quadratic formula and then use $x$ to solve for $y$.