Formal statement that functions intersect

Question

How do I express the following as a mathematical statement with quantification of all variables and making the universe explicit.

"The curves $y=1-x^2$ and $y=3x-2$ intersect"

So far I have $(\exists (x,y)\in \mathbb{R})$

Answer 1

No need to even mention the irrelevant $y$:

$$\exists x \in \mathbb{R}\ s.t. 1-x^2 = 3 x-2$$

Just for "culture," here is a graph confirming there are two solutions:

Answer 2

$$\exists x\in \mathbb{R},\, \exists y \in \mathbb{R},\,\,y=1-x^2 \,\wedge\, y=3x-2$$