but, you wanted an algebraic solution.
now we'll work on that.
you have the 2 equations you started out with.
they are:
x^2 + y^2 = 52 (first equation)
4x^2 + y^2 = 64 (second equation)
we can solve by elimination to find the value of x^2.
once we find the value of x^2, we can then find the value of y^2.
subtract the first equation from the second equation to get:
4x^2 - x^2 + y^2 - y^2 = 64 - 52
simplify to get:
3x^2 = 12
divide both sides by 3 to get:
x^2 = 4
solve for x to get x = plus or minus 2.
those will be the x-coordinates of the intersection points.
now we'll solve for y.
in the first equation, replace x^2 with 4 to get:
x^2 + y^2 = 52 becomes 4 + y^2 = 52
subtract 4 from both sides of this equation to get:
y^2 = 48
take the square root of both sides of this equation to get:
y = plus or minus sqrt(48) = 6.92820323.
your x values are plus or minus 2.
your y values are plus or minus 6.92820323 which is equal to 6.928 rounded to 3 decimal places.
your intersection points will be:
x y
-2 -6.928
-2 6.928
2 -6.928
2 6.928
if you replace any of these point pair combinations in the original equations, those equations will be true.
these same intersection points are shown on the graph.
the algebraic solution is equivalent to the graphical solution.