What are the intercepting points of

y^2=-4x and x^2=y

I set both equations equal to 0, but only get x=0 and there is another intercepting point. please help!

Graphically, y=x^2 is positive for all values of x, whether positive or negative.
The second equation is y positive for the
negative values of X without getting into imaginary numbers. The curves intersect at y=~2.5 and x=~1.6

Analytically, solve y^2=-4X for x, then substitue in Y=x^2. After factoring, you get a cubic equation y^3-16=0. Only one root isn't imaginary: y=2.52 When you substitute that to solve for x, it is -1.59, which checks with the graph.

To find the intercepting points of the given equations, you can approach it both graphically and analytically.

Graphically:
1. Plot the graphs of the equations y^2 = -4x and x^2 = y on the same coordinate system.
2. The first equation, y^2 = -4x, represents a parabola opening to the left. The vertex of this parabola is at the origin (0,0).
3. The second equation, x^2 = y, represents a parabola opening upwards with its vertex also at the origin.
4. By observing the plot, you can see that the curves intersect at approximately y = 2.5 and x = 1.6.

Analytically:
1. Start with the equation y^2 = -4x.
2. Solve this equation for x to get x = -(y^2)/4.
3. Substitute this value of x into the equation x^2 = y to get (-(y^2)/4)^2 = y.
4. Simplify the equation to y^4/16 = y.
5. Multiply both sides by 16 to eliminate the fraction, resulting in y^4 = 16y.
6. Rearrange the equation to y^4 - 16y = 0.
7. Factor out the common factor y to get y(y^3 - 16) = 0.
8. Set each factor equal to zero and solve for y.
- The first factor, y = 0, gives us one intercepting point at y = 0.
- The second factor, y^3 - 16 = 0, represents a cubic equation. Finding the roots of this equation, we find that y = 2.52 is the non-imaginary root.
9. Now, substitute the value of y = 2.52 back into either of the original equations to solve for x. Using x = -(y^2)/4, we have x = -1.59.
10. Thus, the intercepting points of the given equations are approximately (-1.59, 2.52) and (0, 0).

By following either the graphical or analytical approach, you can determine the intercepting points of the equations y^2 = -4x and x^2 = y.