Let ^@\alpha^@ and ^@\beta^@ be the roots of ^@x^2 - 3 x + c = 0 | Math Question and Answer

Answer:

$0$

Step by Step Explanation:

$\alpha \space \& \space \beta$ are the roots of the equation, therefore,
$\alpha \beta = \dfrac{c}{1} = c \space\space .....(1)$
As $\alpha$ is the root of the equation $x^2 - 3 x + c = 0$ ,
$\alpha^2 - 3\alpha + c = 0 \space \space .....(2)$
Also, $-\alpha$ is the root of the equation $x^2 + 3 x - c = 0$ ,
$\begin{align} & (-\alpha)^2 + 3(-\alpha) - c = 0\\ &\implies \alpha^2 - 3\alpha - c = 0 && .....(3) \end{align}$
On subtracting $eq(2)$ by $eq(3)$ , we get,
$\begin{align} & \alpha^2 - 3\alpha - c - (\alpha^2 - 3\alpha +c) = 0 \\ \implies & \alpha^2 - 3\alpha - c - \alpha^2 + 3\alpha - c = 0 \\ \implies & -2c = 0 \\ \implies & c = 0 \end{align}$
By $eq(1)$ , we have,
$\begin{align} & \alpha\beta = c \\ \implies & \alpha\beta = 0 \end{align}$

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