Question 852363
You must want these equations as given ones:
{{{x^2-2y=11}}} and {{{3x^2+y^2=24}}}.
-------- A parabola intersecting a circle.  Yes, MAYBE four solutions for that system; depends....


What method to solve the system, depends on what you know and what you want to use (unless you are directed to use a particular method).   That choice appears to be YOURS.


To not remove the opportunity for you to begin to solve the system, I will begin a couple of steps off the site page and continue some of it on the site page...
A possible equation resulting from a solution process can be {{{highlight_green(y^2-6y+9=0)}}}.  This is a quadratic equation in y, and it is factorable.


Continuing with that factorization,
{{{(y-3)^2=0}}}
{{{highlight(y=3)}}} and this will mean through {{{x^2=11-2y}}}, that {{{highlight(x=-sqrt(5))}}} or {{{highlight(x=sqrt(5))}}}.
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LOOKS LIKE TWO SOLUTIONS, NOT FOUR SOLUTIONS.


{{{graph(300,300,-5,5,-8,8,(1/2)x^2-11/2,-sqrt(24-3x^2),sqrt(24-3x^2))}}}
One reason the circle appears split is because the circle is made of two functions put together on the same graph.
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This graph very obviously shows two points of intersection; NOT four.