Can somebody explains what I did wrong? Find the points (0,0) and (-1,-1). Eq1) x^2=-y, Eq2) x=-y^2
I replace Eq1 in Eq2--> x=(x^2)^2-->x=x^4--> x^4 -x=0-->x(x^3 -1)=0--> x(x+1)(x^2 +x-1)=0
I got x=0, and x=-1. I plug those into the Eq1 I get 0^2=-y and -1^2=-y. I got the points (0,0) and (-1,-1) but I feel I did something wrong. I got the points but what about the (x^2 +x-1). Why is it still around? What should I do with it? Should there be another point?
I did it this way:
x^2 = -y and x = -y^2
sub the 2nd into the first:
(-y^2)^2 = -y
y^4 + y = 0
y(y^3 + 1) = 0
then y = 0 or y^3 = -1 ---> y = -1
in the 2nd,
if y = 0, then x = 0
if y = -1, then x = -(-1)^2 = -1
so there are your (0,0) and (-1,-1)
following yours ......
the 1st into the 2nd
the 1st: x^2 = -y or y = -x^2
2nd: x = -y^2
x = -(-x^2)^2 = -x^4
x^4 + x = 0
x(x^3 + 1) = 0
same pattern as mine when I solved for y
can you see your error? Watch those nasty negatives when you square them
-y^2 means -(y^2)
I got it. Thank You
To find the points of intersection between the two equations, Eq1) x^2=-y and Eq2) x=-y^2, you correctly substituted Eq1 into Eq2 to get x=(x^2)^2. However, there seems to be some confusion in the subsequent steps.
Let's go through the correct steps to find the points of intersection:
1. Start with Eq1: x^2 = -y
2. Substitute Eq1 in Eq2: x = -(-y)^2 = -y^2
3. Set the equations equal to each other: x = x^4
At this point, you correctly rearranged the equation to x^4 - x = 0.
4. Factor the equation: x(x^3 - 1) = 0
This step is where the (x^3 - 1) term comes in. It represents the remaining factor after factoring out x. So, it is still relevant in finding all the possible solutions.
5. Set each factor equal to zero and solve:
a. x = 0
b. x^3 - 1 = 0
For the first factor, x = 0, you substitute it back into Eq1 to find the corresponding y-coordinate:
Eq1: y = -x^2 = -0^2 = 0
So, one point of intersection is (0, 0).
For the second factor, x^3 - 1 = 0, you can solve it separately:
x^3 - 1 = 0
x^3 = 1
This equation represents a cubic equation, and it has complex solutions. The three solutions are:
x = 1 (real solution)
x = -0.5 + 0.866i (complex solution)
x = -0.5 - 0.866i (complex solution)
For each of these solutions, you substitute them back into Eq1 to find the corresponding y-coordinate.
For x = 1: y = -1
For x = -0.5 + 0.866i: y = -0.5 - 0.866i
For x = -0.5 - 0.866i: y = -0.5 + 0.866i
Therefore, the second set of points of intersection are (1, -1), (-0.5 + 0.866i, -0.5 - 0.866i), (-0.5 - 0.866i).
In summary, there are three points of intersection: (0, 0), (1, -1), and (-0.5 + 0.866i, -0.5 - 0.866i).