Find the value of a for which the equations x | vedclass

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Question

(C) Let the common root be $\alpha$. Then we have: $1) \alpha^3 + a\alpha + 1 = 0$ $2) \alpha^4 + a\alpha^2 + 1 = 0$ Multiply equation $(1)$ by $\alph

Vedclass · Accepted Answer

$-2$

Vedclass · Answer

(C) Let the common root be $\alpha$. Then we have: $1) \alpha^3 + a\alpha + 1 = 0$ $2) \alpha^4 + a\alpha^2 + 1 = 0$ Multiply equation $(1)$ by $\alpha$: $\alpha^4 + a\alpha^2 + \alpha = 0$ Subtract equation $(2)$ from this result: $(\alpha^4 + a\alpha^2 + \alpha) - (\alpha^4 + a\alpha^2 + 1) = 0$ $\alpha - 1 = 0 \Rightarrow \alpha = 1$ Since $\alpha = 1$ is a common root,it must satisfy the first equation: $(1)^3 + a(1) + 1 = 0$ $1 + a + 1 = 0$ $a + 2 = 0 \Rightarrow a = -2$

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

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