`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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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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