The roots of the equations 2x^2 - 5x + 1 = 0 | vedclass

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(B) Let the roots of the first equation $2x^2 - 5x + 1 = 0$ be $\alpha$ and $\beta$. Then,$\alpha \beta = \frac{c}{a} = \frac{1}{2}$. Now,consider the

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Reciprocal and of opposite signs

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(B) Let the roots of the first equation $2x^2 - 5x + 1 = 0$ be $\alpha$ and $\beta$. Then,$\alpha \beta = \frac{c}{a} = \frac{1}{2}$. Now,consider the second equation $x^2 + 5x + 2 = 0$. Let its roots be $\gamma$ and $\delta$. Then,$\gamma \delta = \frac{2}{1} = 2$. Comparing the two equations,we observe that the coefficients of the second equation are the reverse of the first,but with a sign change in the middle term. Specifically,if the roots of $ax^2 + bx + c = 0$ are $\alpha, \beta$,then the roots of $cx^2 + bx + a = 0$ are $\frac{1}{\alpha}, \frac{1}{\beta}$. For the equation $x^2 + 5x + 2 = 0$,which is $2x^2 + 5x + 1 = 0$ (if we consider the reciprocal property),the roots are related to the original. Actually,for $2x^2 - 5x + 1 = 0$,the roots are $\alpha, \beta$. For $x^2 + 5x + 2 = 0$,the roots are $-\frac{1}{\alpha}, -\frac{1}{\beta}$. Thus,the roots are reciprocals of each other and have opposite signs.

The roots of the equations $2x^2 - 5x + 1 = 0$ and $x^2 + 5x + 2 = 0$ are:

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