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If alpha,beta are the roots of a x^2+b x...

If `alpha,beta` are the roots of `a x^2+b x+c=0,(a!=0)` and `alpha+delta,beta+delta` are the roots of `A x^2+B x+C=0,(A!=0)` for some constant `delta` then prove that `(b^2-4a c)/(a^2)=(B^2-4A C)/(A^2)`

Text Solution

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Since, `alpha+beta=-b/a,alphabeta=c/a`
`andalpha=delta+beta+delta=-B/A,(alpha+delta)-(beta+delta)=C/A`
Now, `alpha-beta=(alpha+delta)-(beta+delta)`
`implies(alpha-beta)^(2)=[(alpha+delta)-(beta+delta)]^(2)`
`implies(alpha+beta)^(2)-4alphabeta=(bar(beta+delta)+bar(beta+delta))^(2)-4(alpha+delta)(beta+delta)`
`implies(-(b)/(a))^(2)-(4c)/(a)=(-(B)/(A))^(2)-(4C)/(A)`
`implies(b^(2))/(a^(2))-(4c)/(a)=(B^(2))/(A^(2))-(4C)/(A)`
`implies(b^(2)-4ac)/(a^(2))=(B^(2)-4AC)/(A^(2))`
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