If the equations $ax^{2} + bx + c = 0$ and $cx^{2} + bx + a = 0,a,b,c \in R$ and $ac
eq 0$ have $a

Question

If the equations $ax^{2} + bx + c = 0$ and $cx^{2} + bx + a = 0,a,b,c \in R$ and $ac 
eq 0$ have $a$ common non real root, then

Accepted Answer

$D_{1} = b^{2} - 4ac$ and $D_{2} = b^{2} - 4ac$

$$\Rightarrow \ D_{1} < 0\text{~and~}D_{2} < 0$$

⇒ Equation have both roots common

$$\begin{matrix} \Rightarrow & \frac{a}{c} = \frac{b}{b} = \frac{c}{a} \\ \Rightarrow & a = c\text{~and~}b^{2} < 4ac = 4a^{2} \\ & - 2|a| < |b| < 2|a| \end{matrix}$$

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