If the equations
\[
ax^2+2bx+3c=0,\qquad
3x^2+8x+15=0
\]
have a common root, where \(a,b,c\) are sides of triangle \(ABC\), then
\[
\cos2A+\cos2B+\cos2C=
\]
Show Hint
If the sides of a triangle are in the ratio
\[
\boxed{3:4:5,}
\]
then the triangle is right-angled.
Hence,
\[
\boxed{\cos2A+\cos2B+\cos180^\circ=-1.}
\]