Euler’s Method of Proving Infinite Solutions.

Suppose for an equation say – a^2 + b^2 + 3 = ab

For proving Infinite solutions ,

Assume the existence of a non-trivial solution by keeping one of the variables fixed or a constant, Now the equation is in one variable. 

Since the equation is a quadratic , there should exist another root .

Similarly for the other root there exists another root… thus there exists infinite roots.

That is … in

a^2 + b^2 + 3 =ab

If b is a constant ,

a^2 - ab + (b^2+3) = 0

is a quadratic.

Hence we can find two roots.

let roots be \alpha , \beta

Now keep \alpha fixed and b as a variable ,

b^2 - \alpha\cdot b + ( (\alpha)^2 + 3 ) = 0

This again has two roots.

Thus , this way .. the equation has infinite roots.


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