Sketch both the circle $x^2 + y^2 = 25$ and the parabola $y = x^2 - 13$.

1. How many points of intersection are there?

   Hint (a):

   Look at the graph.

   

   Answer (a):

   $4$ points

2. Find the coordinates of these points algebraically.

   Step 1(b):

   Substitute $x^² - 13$ into the first equation.  $x^2 + (x^2 - 13)^2 = 25$

   Step 2(b):

   Solve for $x$.  $x^4 - 25x^2 + 144 = 0$

   Step 3(b):

   Factor the equation.  $(x^² - 16)(x^² - 9)$

   Step 4(b):

   $(x − 4)(x + 4)(x − 3)(x + 3)$

   Step 5(b):

   Now substitute in these $x$ values to find the $y$.

   Partial Answer(b):

   $x = \pm4, \pm3$