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  4. The value of 'a' for which the equations x3 + ax + 1 = 0 and x4 + ax2 + 1=0 have a common root is

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Q. The value of $'a'$ for which the equations $x^3 + ax + 1 = 0$ and $x^4 + ax^2 + 1=0$ have a common root is

Complex Numbers and Quadratic Equations

Solution:

Let $\alpha $ be a common root of
$x^3 + ax + 1 = 0 $ and $x^4 + ax^2 + 1 = 0$
$ \therefore \, \alpha^3 + a \alpha + 1 = 0 $ and $ \alpha^4 + a \alpha^2 + 1 = 0$
$i.e. \, \alpha^4 + a \alpha^2 + \alpha = 0$ and $ \alpha^4 + a \alpha^2 + 1=0$
$ \therefore \, \alpha - 1 = 0$
$\Rightarrow \, \alpha = 1$
$ \therefore \, 1 + a \cdot 1 + 1 = 0$
$\Rightarrow \, a = -2$