Is there a way to find the point of intersection of two quadratic Bezier curves?

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Each Bezier curve have two quadratic parametric equations to express the curve.

Can these be utilised to find the point of intersection with the other curve? If so, how? I am really confused.

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Yes, you can compute the points of intersection.

Several techniques for doing this are described in chapter 7 of these notes. Some of them are algebraic, and some iterative/numerical.

Basically, you have to find the zeros of a polynomial of degree 4. This is to be expected, since there will generally be four points of intersection.