question number 5

5. Let f(x), g(x), and h(x) be the quadratic polynomials having positive leading coefficients and real and distinct roots. If each pair of them has a common root, then find the roots of
f(x) + g(x) + h(x) = 0. 

Dear Student,
Please find below the solution to the asked query:

As each pair has common root, hence let roots of fx be α,βroots of gx be β,γroots of hx be γ,αAs leading coefficient is positive , take it as 1fx=x-αx-βgx=x-βx-γhx=x-γx-αfx+gx+hx=0x-αx-β+x-βx-γ+x-γx-α=0x2-α+βx+αβ+x2-β+γx+βγ+x2-γ+αx+γα=03x2-2α+β+γx+αβ+βγ+γα=0Now apply quadratic formula x=2α+β+γ±4α+β+γ2-4×3×αβ+βγ+γα6=2α+β+γ±2α2+β2+γ2+2αβ+βγ+γα-3αβ+βγ+γα6=α+β+γ±α2+β2+γ2-αβ-βγ-γα3

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