The quadratic equations x^2 - 6x + a = 0 and | vedclass

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(D) Let the common root be $\alpha$. Let the other root of the first equation $x^2 - 6x + a = 0$ be $4\beta$ and the other root of the second equation

Vedclass · Accepted Answer

$2$

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(D) Let the common root be $\alpha$. Let the other root of the first equation $x^2 - 6x + a = 0$ be $4\beta$ and the other root of the second equation $x^2 - cx + 6 = 0$ be $3\beta$,where $\beta$ is an integer.From the properties of roots:For the first equation: $\alpha + 4\beta = 6$ and $\alpha(4\beta) = a$.For the second equation: $\alpha + 3\beta = c$ and $\alpha(3\beta) = 6$.From the second equation,we have $3\alpha\beta = 6$,which implies $\alpha\beta = 2$.Since $\alpha$ and $\beta$ are integers,the possible pairs for $(\alpha, \beta)$ are $(1, 2), (2, 1), (-1, -2), (-2, -1)$.Case $1$: If $\alpha = 2$ and $\beta = 1$,then $\alpha + 4\beta = 2 + 4(1) = 6$. This satisfies the first equation's sum of roots condition.Case $2$: If $\alpha = 1$ and $\beta = 2$,then $\alpha + 4\beta = 1 + 4(2) = 9 
eq 6$.Thus,the common root $\alpha$ is $2$.

The quadratic equations $x^2 - 6x + a = 0$ and $x^2 - cx + 6 = 0$ have one root in common. The other roots of the first and second equations are integers in the ratio $4 : 3$. Then the common root is:

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