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  • Show that if p,q,r,s are real numbers and pr=2(p+s), then at least one of the equations x^2+px+q=0 and x^2+rx+s=0 has real roots .

Show that if p,q,r,s are real numbers and pr=2(p+s), then at least one of the equations x^2+px+q=0 and x^2+rx+s=0 has real roots .

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Solution: Let D_1 andD_2 be the discriminants of the given equation.

x^2+px+q=0 and x^2+rx+s=0 , respectively,

Now,                      D_1+D_2=p^2-4q+r^2-4s

                                                  =p^2+r^2-4(q+s)

                                                  =p^2+r^2-2pr                                 [ecause pr=2(p+s)]

                                                =(p-r)^2geq 0Rightarrow D_1+D_2geq 0     

      Hence, at least one of the equations

                                         x^2+px+q=0 and x^2+rx+s=0 has real root.

Posted by

Deependra Verma

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