First slide

Theory of equations

Question

If ${ax}^{2} + bx + c = 0 and {bx}^{2} + cx + a = 0$ have a common root $a \neq 0$ then $\frac{a^{3} + b^{3} + c^{3}}{abc} =$

Moderate

A

1
B

2
C

3
D

None of these

Solution

The condition for common roots gives
${(bc - a^{2})}^{2} = (ca - b^{2}) (ab - c^{2})$
on simplification gives $a^{3} + b^{3} + c^{3} = 3 abc$

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