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Root of a function

The roots or the zeros of a numerical function $$f$$ form the set of values that solve the equation $$f(x)=0$$.

For instance, the roots of the second order expression $$$f(x) = ax^2 + bx+c$$$ with $$a\ne 0$$ depend on the sign of the discriminant $$$\Delta = b^2 - 4 ac.$$$

  • If $$\Delta\lt 0$$, $$f$$ has no real root.
  • If $$\Delta = 0$$, $$f$$ has a unique zero $$-b / (2a)$$, called a repeated root .
  • If $$\Delta\gt 0$$, $$f$$ has two distinct real roots $$$\displaystyle x_- = \frac{-b-\sqrt{\Delta}}{2a},\quad x_+ = \frac{-b+\sqrt{\Delta}}{2a}.$$$

  • $$x^2+1=0$$ has no real root, because $$\Delta = -4\lt 0 $$.
  • $$x^2-2x+1=0$$ has a repeated root $$x=1$$, because $$\Delta = 0 $$.
  • $$x^2+2x=0$$ has a two real roots, because $$\Delta = 4 \gt 0$$. The roots are $$x_- = (-2-2)/2 = -2$$ and $$x_+=(-2+2)/2 = 0$$.