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Mathematics Question on Complex Numbers and Quadratic Equations

If the difference between the roots of x2+ax−b=0{{x}^{2}}+ax-b=0 is equal to the difference between the roots of x2−px+q=0,{{x}^{2}}-px+q=0, then a2−p2{{a}^{2}}-{{p}^{2}} in terms of bb and qq is

A

−4(b+q)-4(b+q)

B

4(b+q)4(b+q)

C

4(b−q)4(b-q)

D

4(q−b)4(q-b)

Answer

−4(b+q)-4(b+q)

Explanation

Solution

Let α,β\alpha ,\beta are the roots of the equation
x2+ax−b=0{{x}^{2}}+ax-b=0
∴\therefore α+β=−a,  αβ=−b\alpha +\beta =-a,\,\,\alpha \beta =-b
and γ,δ\gamma ,\delta
are the roots of the equation
x2−px+q=0{{x}^{2}}-px+q=0
∴\therefore γ+δ=p, γδ=q\gamma +\delta =p,\,\gamma \delta =q
Given, α−β=γ−δ\alpha -\beta =\gamma -\delta
⇒\Rightarrow (α−β)2=(γ−δ)2{{(\alpha -\beta )}^{2}}={{(\gamma -\delta )}^{2}}
⇒\Rightarrow (α+β)2−4αβ=(γ+δ)2−4γδ{{(\alpha +\beta )}^{2}}-4\alpha \beta ={{(\gamma +\delta )}^{2}}-4\gamma \delta
⇒\Rightarrow a2+4b=p2−4q{{a}^{2}}+4b={{p}^{2}}-4q
⇒\Rightarrow a2−p2=−4(b+q){{a}^{2}}-{{p}^{2}}=-4(b+q)