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Review quadratic equation derivation

abstract:
reference: https://en.wikipedia.org/wiki/Quadratic_formula#Derivations_of_the_formula

step	inference rule	input	feed	output	step validity (as per SymPy)
1	111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			9285928292 $ax^2 + bx + c=0$	no validation is available for declarations
3	111975: divide both sides by number of inputs: 1; feeds: 1; outputs: 1 Divide both sides of Eq.~\\ref{eq:#2} by $#1$; yields Eq.~\\ref{eq:#3}.	9285928292 $ax^2 + bx + c=0$	0002424922 $a$	5958392859 $x^2 + (b/a)x+(c/a)=0$	Algebraic error: LHS diff is pdg0001464*(pdg0001939 - pdg0009139)/pdg0009139, RHS diff is 0
4	111282: subtract X from both sides number of inputs: 1; feeds: 1; outputs: 1 Subtract $#1$ from both sides of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	5958392859 $x^2 + (b/a)x+(c/a)=0$	0006644853 $c/a$	5938459282 $x^2 + (b/a)x=-c/a$	LHS diff is pdg0001464*(-pdg0001939 + pdg0009139)/pdg0009139 RHS diff is 0
5	111530: add X to both sides number of inputs: 1; feeds: 1; outputs: 1 Add $#1$ to both sides of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	5938459282 $x^2 + (b/a)x=-c/a$	0004307451 $(b/(2 a))^2$	5928292841 $x^2 + (b/a)x + (b/(2 a))^2=-c/a + (b/(2 a))^2$	valid
6	111886: change variable X to Y number of inputs: 1; feeds: 2; outputs: 1 Change variable $#1$ to $#2$ in Eq.~\\ref{eq:#3}; yields Eq.~\\ref{eq:#4}.	8582954722 $x^2 + 2 x h + h^2=(x + h)^2$	0000999900 $b/(2 a)$ 0004858592 $h$	5928285821 $x^2 + 2 x (b/(2 a)) + (b/(2 a))^2=(x + (b/(2 a)))^2$	LHS diff is -pdg0001464pdg0001939/pdg0009139 + 2pdg0001464pdg0003410 - pdg00019392/(4pdg00091392) + pdg00034102 RHS diff is (pdg0001464 + pdg0003410)*2 - (2pdg0001464pdg0009139 + pdg0001939)2/(4pdg0009139**2)
7	111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	5928285821 $x^2 + 2 x (b/(2 a)) + (b/(2 a))^2=(x + (b/(2 a)))^2$		5959282914 $x^2 + x(b/a) + (b/(2 a))^2=(x+(b/(2 a)))^2$	valid
7.5	111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			8582954722 $x^2 + 2 x h + h^2=(x + h)^2$	no validation is available for declarations
8	111355: LHS of expr 1 equals LHS of expr 2 number of inputs: 2; feeds: 0; outputs: 1 LHS of Eq.~\\ref{eq:#1} is equal to LHS of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	5959282914 $x^2 + x(b/a) + (b/(2 a))^2=(x+(b/(2 a)))^2$ 5928292841 $x^2 + (b/a)x + (b/(2 a))^2=-c/a + (b/(2 a))^2$		9385938295 $(x+(b/(2 a)))^2=-(c/a) + (b/(2 a))^2$	valid
9	111524: square root both sides number of inputs: 1; feeds: 0; outputs: 2 Take the square root of both sides of Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2} and Eq.~\\ref{eq:#3}.	9385938295 $(x+(b/(2 a)))^2=-(c/a) + (b/(2 a))^2$		9582958294 $x+(b/(2 a))=\sqrt{(b/(2 a))^2 - (c/a)}$ 5982958249 $x+(b/(2 a))=-\sqrt{(b/(2 a))^2 - (c/a)}$	recognized infrule but not yet supported
10	111282: subtract X from both sides number of inputs: 1; feeds: 1; outputs: 1 Subtract $#1$ from both sides of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	5982958249 $x+(b/(2 a))=-\sqrt{(b/(2 a))^2 - (c/a)}$	0002838490 $b/(2 a)$	9582958293 $x=\sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a))$	LHS diff is 0 RHS diff is -sqrt((pdg0001939*2 - 4pdg0004231pdg0009139)/pdg0009139*2)
10.5	111282: subtract X from both sides number of inputs: 1; feeds: 1; outputs: 1 Subtract $#1$ from both sides of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	9582958294 $x+(b/(2 a))=\sqrt{(b/(2 a))^2 - (c/a)}$	0002449291 $b/(2 a)$	5982958248 $x=-\sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a))$	LHS diff is 0 RHS diff is sqrt((pdg0001939*2 - 4pdg0004231pdg0009139)/pdg0009139*2)
11	111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	5982958248 $x=-\sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a))$		9999999968 $x=\frac{-b-\sqrt{b^2-4ac}}{2 a}$	LHS diff is 0 RHS diff is (-pdg0009139sqrt((pdg00019392 - 4pdg0004231pdg0009139)/pdg00091392) + sqrt(pdg00019392 - 4pdg0004231pdg0009139))/(2pdg0009139)
11.5	111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	9582958293 $x=\sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a))$		9999999969 $x=\frac{-b+\sqrt{b^2-4ac}}{2 a}$	LHS diff is 0 RHS diff is (pdg0009139sqrt((pdg00019392 - 4pdg0004231pdg0009139)/pdg00091392) - sqrt(pdg00019392 - 4pdg0004231pdg0009139))/(2pdg0009139)
14	111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\\ref{eq:#1} is one of the final equations.	9999999968 $x=\frac{-b-\sqrt{b^2-4ac}}{2 a}$			no validation is available for declarations
15	111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\\ref{eq:#1} is one of the final equations.	9999999969 $x=\frac{-b+\sqrt{b^2-4ac}}{2 a}$			no validation is available for declarations

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timing of Neo4j queries:

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step	inference rule	input	feed	output	step validity (as per SymPy)
1	111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			9285928292 \(ax^2 + bx + c=0\)	no validation is available for declarations
3	111975: divide both sides by number of inputs: 1; feeds: 1; outputs: 1 Divide both sides of Eq.~\\ref{eq:#2} by $#1$; yields Eq.~\\ref{eq:#3}.	9285928292 \(ax^2 + bx + c=0\)	0002424922 \(a\)	5958392859 \(x^2 + (b/a)x+(c/a)=0\)	Algebraic error: LHS diff is pdg0001464*(pdg0001939 - pdg0009139)/pdg0009139, RHS diff is 0
4	111282: subtract X from both sides number of inputs: 1; feeds: 1; outputs: 1 Subtract $#1$ from both sides of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	5958392859 \(x^2 + (b/a)x+(c/a)=0\)	0006644853 \(c/a\)	5938459282 \(x^2 + (b/a)x=-c/a\)	LHS diff is pdg0001464*(-pdg0001939 + pdg0009139)/pdg0009139 RHS diff is 0
5	111530: add X to both sides number of inputs: 1; feeds: 1; outputs: 1 Add $#1$ to both sides of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	5938459282 \(x^2 + (b/a)x=-c/a\)	0004307451 \((b/(2 a))^2\)	5928292841 \(x^2 + (b/a)x + (b/(2 a))^2=-c/a + (b/(2 a))^2\)	valid
6	111886: change variable X to Y number of inputs: 1; feeds: 2; outputs: 1 Change variable $#1$ to $#2$ in Eq.~\\ref{eq:#3}; yields Eq.~\\ref{eq:#4}.	8582954722 \(x^2 + 2 x h + h^2=(x + h)^2\)	0000999900 \(b/(2 a)\) 0004858592 \(h\)	5928285821 \(x^2 + 2 x (b/(2 a)) + (b/(2 a))^2=(x + (b/(2 a)))^2\)	LHS diff is -pdg0001464pdg0001939/pdg0009139 + 2pdg0001464pdg0003410 - pdg00019392/(4pdg00091392) + pdg00034102 RHS diff is (pdg0001464 + pdg0003410)*2 - (2pdg0001464pdg0009139 + pdg0001939)2/(4pdg0009139**2)
7	111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	5928285821 \(x^2 + 2 x (b/(2 a)) + (b/(2 a))^2=(x + (b/(2 a)))^2\)		5959282914 \(x^2 + x(b/a) + (b/(2 a))^2=(x+(b/(2 a)))^2\)	valid
7.5	111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			8582954722 \(x^2 + 2 x h + h^2=(x + h)^2\)	no validation is available for declarations
8	111355: LHS of expr 1 equals LHS of expr 2 number of inputs: 2; feeds: 0; outputs: 1 LHS of Eq.~\\ref{eq:#1} is equal to LHS of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	5959282914 \(x^2 + x(b/a) + (b/(2 a))^2=(x+(b/(2 a)))^2\) 5928292841 \(x^2 + (b/a)x + (b/(2 a))^2=-c/a + (b/(2 a))^2\)		9385938295 \((x+(b/(2 a)))^2=-(c/a) + (b/(2 a))^2\)	valid
9	111524: square root both sides number of inputs: 1; feeds: 0; outputs: 2 Take the square root of both sides of Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2} and Eq.~\\ref{eq:#3}.	9385938295 \((x+(b/(2 a)))^2=-(c/a) + (b/(2 a))^2\)		9582958294 \(x+(b/(2 a))=\sqrt{(b/(2 a))^2 - (c/a)}\) 5982958249 \(x+(b/(2 a))=-\sqrt{(b/(2 a))^2 - (c/a)}\)	recognized infrule but not yet supported
10	111282: subtract X from both sides number of inputs: 1; feeds: 1; outputs: 1 Subtract $#1$ from both sides of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	5982958249 \(x+(b/(2 a))=-\sqrt{(b/(2 a))^2 - (c/a)}\)	0002838490 \(b/(2 a)\)	9582958293 \(x=\sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a))\)	LHS diff is 0 RHS diff is -sqrt((pdg0001939*2 - 4pdg0004231pdg0009139)/pdg0009139*2)
10.5	111282: subtract X from both sides number of inputs: 1; feeds: 1; outputs: 1 Subtract $#1$ from both sides of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	9582958294 \(x+(b/(2 a))=\sqrt{(b/(2 a))^2 - (c/a)}\)	0002449291 \(b/(2 a)\)	5982958248 \(x=-\sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a))\)	LHS diff is 0 RHS diff is sqrt((pdg0001939*2 - 4pdg0004231pdg0009139)/pdg0009139*2)
11	111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	5982958248 \(x=-\sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a))\)		9999999968 \(x=\frac{-b-\sqrt{b^2-4ac}}{2 a}\)	LHS diff is 0 RHS diff is (-pdg0009139sqrt((pdg00019392 - 4pdg0004231pdg0009139)/pdg00091392) + sqrt(pdg00019392 - 4pdg0004231pdg0009139))/(2pdg0009139)
11.5	111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	9582958293 \(x=\sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a))\)		9999999969 \(x=\frac{-b+\sqrt{b^2-4ac}}{2 a}\)	LHS diff is 0 RHS diff is (pdg0009139sqrt((pdg00019392 - 4pdg0004231pdg0009139)/pdg00091392) - sqrt(pdg00019392 - 4pdg0004231pdg0009139))/(2pdg0009139)
14	111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\\ref{eq:#1} is one of the final equations.	9999999968 \(x=\frac{-b-\sqrt{b^2-4ac}}{2 a}\)			no validation is available for declarations
15	111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\\ref{eq:#1} is one of the final equations.	9999999969 \(x=\frac{-b+\sqrt{b^2-4ac}}{2 a}\)			no validation is available for declarations