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Respective Discriminants Of These Equations If α,β Are The Roots Of Ax^2+Bx+c=0;α+h,β+h Are The Root

Referencing Styles : MLA | Pages : 120

Given, α, β are the roots of equation ax2+bx+c=0.

Hence, α+β = -b/a and αβ = c/a.

If D1 is the discriminant then D1 = b^2- 4ac.

= a^2(b^2/a^2 – 4c/a) = a^2((-b/a)^2 – 4(c/a))

Hence, D1 = a^2[(α+β)^2 - 4αβ] = a^2(α^2 + β^2 -2αβ)

Also, given α+h and β+h are the roots of equation px^2+qx+r=0

Hence, α+h + β+h = -q/p and  (α+h)(β+h) = r/p.

Now, D2 is the discriminant of the second equation then

D2 = q^2 – 4pr = p^2[q^2/p^2 – 4r/p] = p^2[(-q/p)^2 – 4(r/p)]

= p^2[(α+h+β+h)^2 – 4(α+h)(β+h)] = p^2[(α+h – β-h)^2]

So, D2 = p^2(α^2 + β^2 – 2αβ)

Hence, D1:D2 =

Hence, the ratio of two discriminants is the square of ratio between the coefficients of highest power of the two equations.

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