The command RepresentingQuantifierFreeFormula(pbx) returns the representing quantifier-free formula of  the parametric box pbx.

The command RepresentingQuantifierFreeFormula(rsas, R) returns the representing quantifier-free formula of the regular semi-algebraic system rsas.

See the page SemiAlgebraicSetTools for the definition of a regular semi-algebraic system and that of a parametric box.

$\mathrm{with}\left(\mathrm{RegularChains}\right)\:$

$\mathrm{with}\left(\mathrm{ParametricSystemTools}\right)\:$

$\mathrm{with}\left(\mathrm{SemiAlgebraicSetTools}\right)\:$

$R≔\mathrm{PolynomialRing}\left(\left[x\,b\,a\,c\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$F≔\left[a{x}^2\+bx\+c\right]$

$F≔\left[a{x}^2+bx+c\right]$

$N≔\left[\right]$

$N≔\left[\right]$

$P≔\left[x\right]$

$P≔\left[x\right]$

$H≔\left[a\right]$

$H≔\left[a\right]$

$\mathrm{rrc}≔\mathrm{RealRootClassification}\left(F\,\left[\right]\,\left[x\right]\,\left[a\right]\,3\,2\,R\right)$

$\mathrm{rrc}≔\left[\left[\mathrm{regular\_semi\_algebraic\_set}\right]\,\mathrm{border\_polynomial}\right]$

$\mathrm{rsas}≔{{\mathrm{rrc}}_1}_1$

$\mathrm{rsas}≔\mathrm{regular\_semi\_algebraic\_set}$

$\mathrm{pbx}≔\mathrm{RepresentingBox}\left(\mathrm{rsas}\,R\right)$

$\mathrm{pbx}≔\mathrm{parametric\_box}$

$\mathrm{IsParametricBox}\left(\mathrm{pbx}\right)$

$\mathrm{true}$

$\mathrm{qff}≔\mathrm{RepresentingQuantifierFreeFormula}\left(\mathrm{pbx}\right)$

$\mathrm{qff}≔\mathrm{quantifier\_free\_formula}$

$\mathrm{Info}\left(\mathrm{qff}\,R\right)$

$\left[\left[c\,a\,b\,4ac-b^2\right]\,\left[\left[\mathrm{-1}\,\mathrm{-1}\,1\,\mathrm{-1}\right]\,\left[1\,1\,\mathrm{-1}\,\mathrm{-1}\right]\right]\right]$

$F≔\left[a{x}^2\&plus;bx\&plus;c\&equals;0\&comma;0<x\&comma;a\ne 0\right]$

$F≔\left[a{x}^2+bx+c=0\&comma;0<x\&comma;a\ne 0\right]$

$R≔\mathrm{PolynomialRing}\left(\left[x\&comma;c\&comma;b\&comma;a\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$\mathrm{out}≔\mathrm{LazyRealTriangularize}\left(F\&comma;R\&comma;\mathrm{output}\&equals;\mathrm{list}\right)$

$\mathrm{out}≔\left[\mathrm{regular\_semi\_algebraic\_system}\right]$

$\mathrm{map}\left(\mathrm{Display}\&comma;\mathrm{out}\&comma;R\right)$

$\left[\left\{\begin{array}{cc}a{x}^2+bx+c=0& \\ x>0& \\ \left\{\begin{array}{cc}-4ca+b^2>0\mathbf{and}b<0\mathbf{and}c>0\mathbf{and}a\ne 0& \\ \mathbf{or}-4ca+b^2>0\mathbf{and}b>0\mathbf{and}c>0\mathbf{and}a<0& \\ \mathbf{or}-4ca+b^2>0\mathbf{and}b>0\mathbf{and}c<0\mathbf{and}a\ne 0& \\ \mathbf{or}-4ca+b^2>0\mathbf{and}b<0\mathbf{and}c<0\mathbf{and}a>0& \end{array}\right.& \end{array}\right.\right]$

$P≔\mathrm{PositiveInequalities}\left({\mathrm{out}}_1\&comma;R\right)$

$P≔\left[x\right]$

$\mathrm{rc}≔\mathrm{RepresentingChain}\left({\mathrm{out}}_1\&comma;R\right)\&semi;$$\mathrm{Display}\left(\mathrm{rc}\&comma;R\right)$

$\mathrm{rc}≔\mathrm{regular\_chain}$

$\left\{\begin{array}{cc}a{x}^2+bx+c=0& \\ a\ne 0& \end{array}\right.$

$\mathrm{qff}≔\mathrm{RepresentingQuantifierFreeFormula}\left({\mathrm{out}}_1\right)\&semi;$$\mathrm{Display}\left(\mathrm{qff}\&comma;R\right)$

$\mathrm{qff}≔\mathrm{quantifier\_free\_formula}$

$-4ca+b^2>0\mathbf{and}b<0\mathbf{and}c>0\mathbf{and}a\ne 0$

$\mathbf{or}-4ca+b^2>0\mathbf{and}b>0\mathbf{and}c>0\mathbf{and}a<0$

$\mathbf{or}-4ca+b^2>0\mathbf{and}b>0\mathbf{and}c<0\mathbf{and}a\ne 0$

$\mathbf{or}-4ca+b^2>0\mathbf{and}b<0\mathbf{and}c<0\mathbf{and}a>0$

$\mathrm{Display}\left({\mathrm{out}}_1\&comma;R\right)$

$\left\{\begin{array}{cc}a{x}^2+bx+c=0& \\ x>0& \\ \left\{\begin{array}{cc}-4ca+b^2>0\mathbf{and}b<0\mathbf{and}c>0\mathbf{and}a\ne 0& \\ \mathbf{or}-4ca+b^2>0\mathbf{and}b>0\mathbf{and}c>0\mathbf{and}a<0& \\ \mathbf{or}-4ca+b^2>0\mathbf{and}b>0\mathbf{and}c<0\mathbf{and}a\ne 0& \\ \mathbf{or}-4ca+b^2>0\mathbf{and}b<0\mathbf{and}c<0\mathbf{and}a>0& \end{array}\right.& \end{array}\right.$