Given that equations x2 + 2x + 3 = 0 and
ax2 + bx + c = 0 have a common root.
Root of x2 + 2x + 3 = 0 is
x = \(\frac{-2\pm\sqrt{4-4\times1\times3}}2\)
= \(\frac{-2\pm\sqrt{4-12}}2\)
= \(\frac{-2\pm2\sqrt2i}2\) = -1\(\pm\)√2i
Since, roots of x2 + 2x + 3 = 0 are imaginary and it is given that one root is common with ax2 + bx + c = 0
\(\therefore\) Both roots are common of equations x2 + 2x + 3 = 0
& ax2 + bx + c = 0 (Both roots are conjugate to each other)
\(\therefore\) x2 + 2x + 3 = 0 and x2 + b/a x + c/a = 0
will represent same equation.
\(\therefore\) b/a = 2 & c/a = 3
a : b : c = a : 2a : 3a = 1 : 2 : 3.