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Review quadratic equation derivation

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.
  1. 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}.
  1. 9285928292
    \(ax^2 + bx + c=0\)
  1. 0002424922
    \(a\)
  1. 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}.
  1. 5958392859
    \(x^2 + (b/a)x+(c/a)=0\)
  1. 0006644853
    \(c/a\)
  1. 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}.
  1. 5938459282
    \(x^2 + (b/a)x=-c/a\)
  1. 0004307451
    \((b/(2 a))^2\)
  1. 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}.
  1. 8582954722
    \(x^2 + 2 x h + h^2=(x + h)^2\)
  1. 0000999900
    \(b/(2 a)\)
  1. 0004858592
    \(h\)
  1. 5928285821
    \(x^2 + 2 x (b/(2 a)) + (b/(2 a))^2=(x + (b/(2 a)))^2\)
 LHS diff is -pdg0001464*pdg0001939/pdg0009139 + 2*pdg0001464*pdg0003410 - pdg0001939**2/(4*pdg0009139**2) + pdg0003410**2 RHS diff is (pdg0001464 + pdg0003410)**2 - (2*pdg0001464*pdg0009139 + pdg0001939)**2/(4*pdg0009139**2)
7
  • 111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.
  1. 5928285821
    \(x^2 + 2 x (b/(2 a)) + (b/(2 a))^2=(x + (b/(2 a)))^2\)
  1. 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.
  1. 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}.
  1. 5959282914
    \(x^2 + x(b/a) + (b/(2 a))^2=(x+(b/(2 a)))^2\)
  1. 5928292841
    \(x^2 + (b/a)x + (b/(2 a))^2=-c/a + (b/(2 a))^2\)
  1. 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}.
  1. 9385938295
    \((x+(b/(2 a)))^2=-(c/a) + (b/(2 a))^2\)
  1. 9582958294
    \(x+(b/(2 a))=\sqrt{(b/(2 a))^2 - (c/a)}\)
  1. 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}.
  1. 5982958249
    \(x+(b/(2 a))=-\sqrt{(b/(2 a))^2 - (c/a)}\)
  1. 0002838490
    \(b/(2 a)\)
  1. 9582958293
    \(x=\sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a))\)
 LHS diff is 0 RHS diff is -sqrt((pdg0001939**2 - 4*pdg0004231*pdg0009139)/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}.
  1. 9582958294
    \(x+(b/(2 a))=\sqrt{(b/(2 a))^2 - (c/a)}\)
  1. 0002449291
    \(b/(2 a)\)
  1. 5982958248
    \(x=-\sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a))\)
 LHS diff is 0 RHS diff is sqrt((pdg0001939**2 - 4*pdg0004231*pdg0009139)/pdg0009139**2)
11
  • 111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.
  1. 5982958248
    \(x=-\sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a))\)
  1. 9999999968
    \(x=\frac{-b-\sqrt{b^2-4ac}}{2 a}\)
 LHS diff is 0 RHS diff is (-pdg0009139*sqrt((pdg0001939**2 - 4*pdg0004231*pdg0009139)/pdg0009139**2) + sqrt(pdg0001939**2 - 4*pdg0004231*pdg0009139))/(2*pdg0009139)
11.5
  • 111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.
  1. 9582958293
    \(x=\sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a))\)
  1. 9999999969
    \(x=\frac{-b+\sqrt{b^2-4ac}}{2 a}\)
 LHS diff is 0 RHS diff is (pdg0009139*sqrt((pdg0001939**2 - 4*pdg0004231*pdg0009139)/pdg0009139**2) - sqrt(pdg0001939**2 - 4*pdg0004231*pdg0009139))/(2*pdg0009139)
14
  • 111341: declare final expression
  • number of inputs: 1; feeds: 0; outputs: 0
  • Eq.~\\ref{eq:#1} is one of the final equations.
  1. 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.
  1. 9999999969
    \(x=\frac{-b+\sqrt{b^2-4ac}}{2 a}\)
 no validation is available for declarations


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