What's wrong in my thinking about Bézout's theorem? First, I know that every hypersurface of degree $d$ defined in $\mathbb{CP}^n$ is diffeomorphic. By using this fact, I wanted to calculate the Euler characteristic of hypersurface of degree $d$.
To begin, let $\chi_n^d$ be an Euler characteristic of projective complex hypersurface of degree $d$. In case of $n=2$, from $\chi(A \cup B)=\chi(A)+\chi(B)-\chi(A \cap B)$ and Bézout's theorem, I derived $\chi^{d_1+d_2}_2=\chi^{d_1}_2+\chi^{d_2}_2-d_1d_2$. However, from the genus-degree formula, we know that $\chi^{d}_2=d(3-d)$, and it does not fit in my relation.
What's wrong in my argument? And, is there any alternative way to calculate $\chi^d_n$ by using its equation? From the above property, I can choose any equation for smooth hypersurface so I taught I could calculate easily at first.... but now, it is not.

The correct statement is that every smooth hypersurface of degree $d$ in $\mathbb{CP}^n$ is diffeomorphic. The union of a smooth hypersurface of degree $d_1$ and a smooth hypersurface of degree $d_2$ is a singular hypersurface of degree $d_1 + d_2$. 
Consider in particular the case $n = 2, d_1 = d_2 = 1$. A smooth hypersurface of degree either $1$ or $2$ is just $\mathbb{CP}^1$, which has Euler characteristic $2$. A singular hypersurface of degree $2$ obtained from the union of two smooth hypersurfaces of degree $1$ is two copies of $\mathbb{CP}^1$ stuck together at a point, which has Euler characteristic $3$. 
You can calculate the Euler characteristic of a smooth hypersurface using Chern classes. See this blog post for details. 
