If the difference between the roots of the eq | vedclass

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(D) Let $\alpha, \beta$ be the roots of $x^2 - bx + c = 0$ and $\alpha', \beta'$ be the roots of $x^2 - cx + b = 0$.The difference between the roots o

Vedclass · Accepted Answer

$-4$

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(D) Let $\alpha, \beta$ be the roots of $x^2 - bx + c = 0$ and $\alpha', \beta'$ be the roots of $x^2 - cx + b = 0$.The difference between the roots of $x^2 - bx + c = 0$ is $|\alpha - \beta| = \sqrt{(\alpha + \beta)^2 - 4\alpha\beta} = \sqrt{b^2 - 4c}$.The difference between the roots of $x^2 - cx + b = 0$ is $|\alpha' - \beta'| = \sqrt{(\alpha' + \beta')^2 - 4\alpha'\beta'} = \sqrt{c^2 - 4b}$.Given that the differences are equal:$\sqrt{b^2 - 4c} = \sqrt{c^2 - 4b}$Squaring both sides:$b^2 - 4c = c^2 - 4b$$b^2 - c^2 = 4c - 4b$$(b - c)(b + c) = -4(b - c)$Assuming $b 
eq c$ (if $b = c$,the equations are identical,which is a trivial case),we divide by $(b - c)$:$b + c = -4$.

If the difference between the roots of the equations $x^2 - bx + c = 0$ and $x^2 - cx + b = 0$ is the same,then $b + c = \dots$

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