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Let abc be sides of triangle . No two are equal and € belongs to real numbers.If roots of x.x +2(a+b+c)x+3€(ab+bc+ca)=0 are real. Then range of€=?
For roots to be real its Discriminant should be greater than equal to zero.It should not be imaginary.Hence,4(a+b+c)^2 – 12€(ab+bc+ca) greater than equal to zero.
In triangles we have the inequality which can be written as The left inequality arises from and the right inequality from the triangle inequalities squared i.e. and summing up.If then the quadratic has real roots So, we need if it has to be true for any value of a,b,c Also, if then it surely has imaginary roots. For it is ambiguous whether the roots are real or imaginary.
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