Question 1177234
using the Descartes' Rule of Signs

{{{x^2 -2x -1}}}....look for changes in sine

So, the coefficients are {{{1}}},{{{-2}}},{{{-1}}}.

As can be seen, there is {{{1}}} change.

This means that there is {{{1}}} positive real root.


Replace {{{x}}} by {{{(-x)}}}:

 {{{(-x)^2 - 2(-x) -1=x^2 + 2x -1}}}

The coefficients are {{{1}}},{{{2}}},{{{-1}}}.

As can be seen, there is {{{1}}} change.

This means that there is {{{1 }}}negative real root.


 The discriminant determines the nature of the roots of a quadratic equation. The word 'nature' refers to the types of numbers the roots can be — namely real, rational, irrational or imaginary.

 If the discriminant is positive, we know that we have {{{2}}} {{{real}}} solutions that could be both positive, both negative, or one solution is positive and other is negative 


so, {{{b^2-4ac=(-2)^2-4*1(-1)=8}}}=>discriminant is positive, we know that we have {{{2}}} {{{real}}} solutions