Question

$\huge\rm\underline\purple{Question :-}$

If Alpha and Beta are the roots of ax + bx + c then find the expression whose roots are (1/alpha) and (1/Beta).



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Answer 1

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Answer 2

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If Alpha and Beta are the roots of ax + bx + c then find the expression whose roots are (1/alpha) and (1/Beta) ?

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$We \: know \: that \: α+β=− \frac{b}{a} \: and \: αβ= \frac{c}{a} \\ The \: sum \: of \: the \: new \: roots \: is \: α+ \frac{1}{β} +β+ \frac{1}{a} =− \frac{b}{a} + \frac{α+β}{αβ} =− \frac{b}{a} - \frac{b}{a} \\ + \frac{c}{a} =− \frac{b}{a} − \frac{b}{c} = − \frac{bc + ab}{ac} \\ The \: product \: of \: the \: new \: roots \: is \\ (a + \frac{1}{β} ) \: \: \: (β + \frac{1}{a} ) \: = aβ + 2 + \frac{1}{aβ} = 2 \\ + \frac{c}{a} + \frac{a}{c} = \frac{2ac + a² + c²}{ac} = \frac{(a + c)²}{ac} \\ The \: new \: equation \: is \\ x² + \frac{bc + ab}{ac} x + \frac {(a + c)² }{ac} = 0 => acx² \\ + b (a + c) x + (a + c)² = 0$