If the difference between the roots of x^2+ax | vedclass

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(A) Let $\alpha_1, \beta_1$ be the roots of $x^2+ax+b=0$. The difference of roots is $|\alpha_1 - \beta_1| = \sqrt{a^2-4b}$.Let $\alpha_2, \beta_2$ be

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$a+b+4=0$

Vedclass · Answer

(A) Let $\alpha_1, \beta_1$ be the roots of $x^2+ax+b=0$. The difference of roots is $|\alpha_1 - \beta_1| = \sqrt{a^2-4b}$.Let $\alpha_2, \beta_2$ be the roots of $x^2+bx+a=0$. The difference of roots is $|\alpha_2 - \beta_2| = \sqrt{b^2-4a}$.Given that the differences are equal,we have $\sqrt{a^2-4b} = \sqrt{b^2-4a}$.Squaring both sides,we get $a^2-4b = b^2-4a$.Rearranging the terms,$a^2-b^2 = 4b-4a$.$(a-b)(a+b) = -4(a-b)$.Since $a 
eq b$,we can divide by $(a-b)$,resulting in $a+b = -4$,or $a+b+4=0$.

If the difference between the roots of $x^2+ax+b=0$ and $x^2+bx+a=0$ is the same,and $a \neq b$,then:

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