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`alpha` and `beta` are roots of the equation x^2-px+q=0 .Find the equation whose roots...

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roshan-rox | (Level 1) Valedictorian

Posted June 3, 2013 at 2:18 PM via web

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`alpha` and `beta` are roots of the equation x^2-px+q=0 .Find the equation whose roots are `alpha` (`alpha` +`beta` ) and `beta` (`alpha` +`beta` ).

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jeew-m | College Teacher | (Level 1) Educator Emeritus

Posted June 3, 2013 at 2:34 PM (Answer #1)

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If a quadratic equation `ax^2+bx+c = 0` has roots `r1` and `r2` then we can say;

`r1+r2 = (-b)/a`

`r1xxr2 = c/a`

 

Similarly for `x^2-px+q=0` we can write;

`alpha+beta = -p`

`alphaxxbeta = q`

 

If the quadratic equation which has roots `alpha(alpha+beta)` and `beta(alpha+beta)` represent by `x^2+mx+n = 0` then we can say;

`-m = alpha(alpha+beta)+beta(alpha+beta)`

`n = alpha(alpha+beta)xxbeta(alpha+beta)`

 

`alpha(alpha+beta)+beta(alpha+beta)`

`= (alpha+beta)(alpha+beta)`

`= (alpha+beta)^2`

`= (-p)^2`

`= p^2`

Hence;

`-m = p^2 `

`m = -p^2`

 

`alpha(alpha+beta)xxbeta(alpha+beta)`

`= (alpha+beta)^2xx(alphaxxbeta)`

`= (-p)^2xx(q)`

`= p^2q`

 

Hence;

`n = p^2q`

 

So the quadratic equation that has roots as `alpha(alpha+beta)` and `beta(alpha+beta)` can be given by;

`x^2-p^2x+p^2q = 0`

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