If $a x^{2} + b x + c = 0 a n d b x^{2} + c x + a = 0$ have a common root and a, b, c are non-zero real numbers, then find the value of $\frac{a^{3} + b^{3} + c^{3}}{a b c}$

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Solution

The correct option is B
3

$a x^{2} + b x + c = 0$

$b x^{2} + c x + a = 0$

Let $α$ be the common root,

$\frac{α^{2}}{a b - c^{2}} = \frac{α}{b c - a^{2}} = \frac{1}{a c - b^{2}}$

$\Rightarrow α =$ $\frac{a b - c^{2}}{b c - a^{2}} = \frac{b c - a^{2}}{a c - b^{2}}$

$\Rightarrow$ $a^{2} b c - a c^{3} - a b^{3} + b^{2} c^{2} = b^{2} c^{2} - 2 a^{2} b c + a^{4}$

$\Rightarrow$ $a c^{3} + a^{4} + a b^{3} = 3 a^{2} b c$

$\Rightarrow$ $\frac{a^{3} + b^{3} + c^{3}}{a b c} = 3$

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If ax2+bx+c=0 and bx2+cx+a=0 have a common root and a, b, c are non-zero real numbers, then find the value of a3+b3+c3abc

The correct option is B 3 ax2+bx+c=0 bx2+cx+a=0 Let α be the common root, α2ab−c2=αbc−a2=1ac−b2 ⇒ α= ab−c2bc−a2=bc−a2ac−b2 ⇒a2bc−ac3−ab3+b2c2=b2c2−2a2bc+a4 ⇒ ac3+a4+ab3=3a2bc ⇒a3+b3+c3abc=3

If $a x^{2} + b x + c = 0 a n d b x^{2} + c x + a = 0$ have a common root and a, b, c are non-zero real numbers, then find the value of $\frac{a^{3} + b^{3} + c^{3}}{a b c}$