4. If the equations $x^{2}+2 a x+b^{2}=0$ and $x^{2}+2 c x-b^{2}=0$ have a common root, and $a$, $b$, and $c$ are the sides of a triangle, then this triangle is a $\qquad$-angled triangle.
4. Right-angled Triangle.

Let $x_{0}$ be the common root of the equations $x^{2}+2 a x+b^{2}=0$ and $x^{2}+2 c x-b^{2}=0$, then $x_{0}^{2}+2 a x_{0}+b^{2}=0$, $x_{0}^{2}+2 c x_{0}-b^{2}=0$. Subtracting the two equations yields $x_{0}=\frac{b^{2}}{c-a}$, substituting into the equation $x^{2}+2 a x+b^{2}=0,\left(\frac{b^{2}}{c-a}\right)^{2}+2 a$. $\frac{b^{2}}{c-a}+b^{2}=0 \Rightarrow b^{2}+2 a(c-a)+(c-a)^{2}=0 \Rightarrow b^{2}+c^{2}=a^{2} \Rightarrow$ the triangle is a right-angled triangle.