If ax² + bx + c = 0 and bx² + cx + a = 0 have a common root a ≠ 0 then \(\frac{a^3\ +\ b^3\ +\ c^3}{abc}\) equal to

Concept:

1. Condition for one and two common roots:

If both roots are common, then the condition is

\(\frac{a_1}{a_2}\ =\ \frac{b_1}{b_2}\ =\ \frac{c_1}{c_2}\)

2. a3 + b3 + c3 - 3abc = (a + b + c)(a2 + b2 + c2 - ab - bc - ca)

If a = b = c, then

a3 + b3 + c3 = 3abc

Calculation:

Given that, 

ax2 + bx + c = 0    ----(1)

bx2 + cx + a = 0    ----(2)

According to the question, both roots of equations (1) and (2) are common. Therefore, using the concept discussed above

\(\frac{a}{b}\ =\ \frac{b}{c}\ =\ \frac{c}{a}\)

This will be possible only if

⇒ a = b, b = c, c = a

⇒ a3 + b3 + c3 = 3abc

⇒ \(\frac{a^3\ +\ b^3\ +\ c^3}{abc}\) = 3