First slide
Theory of equations

Given that α, γ are roots of the equation Ax24x+1=0, and β, δ the roots of the equation of Bx26x+1=0, such that α,β,γ, and δ are in H.P., then


Since α,β,γ,δ are in H.P., 1/α,1/β,1/γ,1/δ are in A.P. and they maybe taken as a3d,ad,a+d,a+3d. Replacing x by 1/x, we get the equation whose roots are 1/α,1/β,1/γ,1/δ. Therefore, equation x24x+A=0 has roots a - 3d, a + d and equation x26x+B=0 has roots a - d, a + 3d. Sum of the roots is

    2(ad)=4,2(a+d)=6     a=5/2,d=1/2

Product of the roots is


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