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Question

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

Vedclass · Accepted Answer

$a + b + 4 = 0$

Vedclass · Answer

(A) Let $\alpha_1, \beta_1$ be the roots of the equation $x^2 + ax + b = 0$. The difference of the roots is $|\alpha_1 - \beta_1| = \sqrt{a^2 - 4b}$.Let $\alpha_2, \beta_2$ be the roots of the equation $x^2 + bx + a = 0$. The difference of the 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,we get $a^2 - b^2 + 4a - 4b = 0$.Factoring the expression,we get $(a - b)(a + b) + 4(a - b) = 0$.Since $a 
eq b$,we can divide by $(a - b)$,which gives $(a + b) + 4 = 0$,or $a + b + 4 = 0$.

If the difference between the corresponding 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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