Question

If the roots of the equation l(m-n)x2+m(n-l)x+n(l-m)=0 are equal then prove m=2ln/l+n

Answer 1

Consider the given quadratic equation:



Since the roots of the above equation are equal, the disrciminant is zero.

If is the general quadratic equation, then the roots of the quadratic equation are equal, if 

In our problem,



Thus, the discriminant is



Since the discriminant is zero,



The above equation is a quadratic equation in m which has two roots, say 

Thus sum of the roots, is given by



Therefore,

......(3)

Let us find the product of the roots of the quadratic equation (2).

Thus,

..........(4)

We know that,

.......(5)

Substitute the values of  from equations (3) and (4) and substitute those values in equation (5),

 we have





Since , we have, .

Thus the roots of the equation (2) are equal.

Now consider the sum of roots of the equation:



Since , say m, we have



Answer 2

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