# 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 ,

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

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.