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Let f(x),g(x),a n dh(x) be the quadratic...

Let `f(x),g(x),a n dh(x)` be the quadratic polynomials having positive leading coefficients and real and distinct roots. I eco pair of them has a common roots, then fine the roots of `f(x)+g(x)+h(x)=0.`

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Let `f(x) = a_(1) (x - alpha) (x - beta),`
` g(x) = a_(2) )x - beta) (x - gamma),`
and `h (x) = a_(3)(x - gamma) (x - alpha),`
where `a_(1), a_(2), a_(3)` are positive.
Let `f(x) + g(x) + h(x) = F(x)`
`rArr F(alpha) = a_(2)(alpha - beta)(alpha - gamma)`
`F(beta) = a_(3) (beta - gamma)(beta - alpha)`
`f(gamma) = a_(1) (gamma - alpha)(gamma - beta)`
`rArr F(alpha) F (beta) F(gamma) = - a_(1) a_(2) a _(3) (alpha - beta)^(2) (beta - gamma)^(2) (gamma - alpha )^(2) lt 0 `
So, roots of `F(x) = 0 are and distinct.
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