For what value of a do the equations x^2 - 3x | vedclass

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(D) Let the common root be $\alpha$. Then,$\alpha^2 - 3\alpha + a = 0$ and $\alpha^2 + a\alpha - 3 = 0$.Subtracting the two equations: $(\alpha^2 - 3\

Vedclass · Accepted Answer

$2$

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(D) Let the common root be $\alpha$. Then,$\alpha^2 - 3\alpha + a = 0$ and $\alpha^2 + a\alpha - 3 = 0$.Subtracting the two equations: $(\alpha^2 - 3\alpha + a) - (\alpha^2 + a\alpha - 3) = 0$.This simplifies to: $-3\alpha - a\alpha + a + 3 = 0$.$-\alpha(3 + a) + (a + 3) = 0$.$(a + 3)(1 - \alpha) = 0$.This implies $a = -3$ or $\alpha = 1$.If $a = -3$,the equations become $x^2 - 3x - 3 = 0$ and $x^2 - 3x - 3 = 0$,which are identical and have roots,but the question implies a specific value for $a$ where a common root exists. If $\alpha = 1$ is a root,then $1^2 - 3(1) + a = 0$.$1 - 3 + a = 0$.$a - 2 = 0 \implies a = 2$.

For what value of $a$ do the equations $x^2 - 3x + a = 0$ and $x^2 + ax - 3 = 0$ have a common root?

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