SOLUTION: {{{Find the points (if any) where the circle x^2 + y^2 = 50 intersects the parabola y=x^2-1. Give exact answers. Do not approximate irrational numbers with decimals.}}} I am ab

Algebra ->  Coordinate-system -> SOLUTION: {{{Find the points (if any) where the circle x^2 + y^2 = 50 intersects the parabola y=x^2-1. Give exact answers. Do not approximate irrational numbers with decimals.}}} I am ab      Log On


   

Question 969308: I am able to get a sketch of the graph, but answer comes up in decimals and not exact answers. I am getting points :
(-2.741867, 6.51783441), (2.741867, 6.51783441) Am I close or way off?
Found 2 solutions by Fombitz, Alan3354:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E2%2By%5E2=50
y=x%5E2-1
Substitute,
x%5E2=y%2B1
So,
y%2B1%2By%5E2=50
y%5E2%2By-49=0
y%5E2%2By%2B1%2F4=49%2B1%2F4
%28y%2B1%2F2%29%5E2=197%2F4
y%2B1%2F2=0+%2B-+sqrt%28197%29%2F2
y=-1%2F2+%2B-+sqrt%28197%29%2F2
Then you can calculate x,
x%5E2=1%2F2+%2B-+sqrt%28197%29%2F2
x=0+%2B-+sqrt%281%2F2+%2B-+sqrt%28197%29%2F2%29
Yes, that's it.

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Find the points (if any) where the circle x^2 + y^2 = 50 intersects the parabola y=x^2-1
-----------------
Sub for y in the circle's equation.
x^2 + (x^2-1)^2 = 50
x%5E4+-+x%5E2+%2B+1+=+50
x%5E4+-+x%5E2+-+49+=+0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-1x%2B-49+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-1%29%5E2-4%2A1%2A-49=197.

Discriminant d=197 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--1%2B-sqrt%28+197+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-1%29%2Bsqrt%28+197+%29%29%2F2%5C1+=+7.5178344238091
x%5B2%5D+=+%28-%28-1%29-sqrt%28+197+%29%29%2F2%5C1+=+-6.5178344238091

Quadratic expression 1x%5E2%2B-1x%2B-49 can be factored:
1x%5E2%2B-1x%2B-49+=+%28x-7.5178344238091%29%2A%28x--6.5178344238091%29
Again, the answer is: 7.5178344238091, -6.5178344238091. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-1%2Ax%2B-49+%29

===========
x%5E2+=+1%2F2+%2B-+sqrt%28197%29%2F2
x%5E2+=+%281+%2B-+2sqrt%28197%29%29%2F4
----
x+=+%2Bsqrt%281+%2B+2sqrt%28197%29%29%2F2 real number
x+=+-sqrt%281+%2B+2sqrt%28197%29%29%2F2 real number
x+=+%2Bsqrt%281+-+2sqrt%28197%29%29%2F2 complex number
x+=+-sqrt%281+-+2sqrt%28197%29%29%2F2 complex number
=============
Find y using x%5E2+=+1%2F2+%2B+sqrt%28197%29%2F2
y+=+-1%2F2+%2B+sqrt%28197%29%2F2 for both real number solutions.