If the difference between the roots of the eq | vedclass

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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}| = \sqr

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,we get $(a - b)(a + b) + 4(a - b) = 0$.Since $a 
e b$,we can divide by $(a - b)$ to get $a + b + 4 = 0$.

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

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