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review derivation: quadratic equation derivation

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Notes for this derivation:
https://en.wikipedia.org/wiki/Quadratic_formula#Derivations_of_the_formula

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
10 subtract X from both sides
  1. 5982958249; locally 6608123:
    \(x+(b/(2 a)) = -\sqrt{(b/(2 a))^2 - (c/a)}\)
    \(pdg_{1464} + \frac{pdg_{1939}}{2 pdg_{9139}} = - \sqrt{\frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} - \frac{pdg_{4231}}{pdg_{9139}}}\)
  1. 0002838490:
    \(b/(2 a)\)
    \(\frac{pdg_{1939}}{2 pdg_{9139}}\)
  1. 9582958293; locally 4433112:
    \(x = \sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a))\)
    \(pdg_{1464} = - \frac{pdg_{1939}}{2 pdg_{9139}} + \sqrt{\frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} - \frac{pdg_{4231}}{pdg_{9139}}}\)
LHS diff is 0 RHS diff is -sqrt((pdg1939**2 - 4*pdg4231*pdg9139)/pdg9139**2) 5982958249:
9582958293:
5982958249:
9582958293:
10.5 subtract X from both sides
  1. 9582958294; locally 6608102:
    \(x+(b/(2 a)) = \sqrt{(b/(2 a))^2 - (c/a)}\)
    \(pdg_{1464} + \frac{pdg_{1939}}{2 pdg_{9139}} = \sqrt{\frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} - \frac{pdg_{4231}}{pdg_{9139}}}\)
  1. 0002449291:
    \(b/(2 a)\)
    \(\frac{pdg_{1939}}{2 pdg_{9139}}\)
  1. 5982958248; locally 2657355:
    \(x = -\sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a))\)
    \(pdg_{1464} = - \frac{pdg_{1939}}{2 pdg_{9139}} - \sqrt{\frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} - \frac{pdg_{4231}}{pdg_{9139}}}\)
LHS diff is 0 RHS diff is sqrt((pdg1939**2 - 4*pdg4231*pdg9139)/pdg9139**2) 9582958294:
5982958248:
9582958294:
5982958248:
11 simplify
  1. 5982958248; locally 2657355:
    \(x = -\sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a))\)
    \(pdg_{1464} = - \frac{pdg_{1939}}{2 pdg_{9139}} - \sqrt{\frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} - \frac{pdg_{4231}}{pdg_{9139}}}\)
  1. 9999999968; locally 8811221:
    \(x = \frac{-b-\sqrt{b^2-4ac}}{2 a}\)
    \(pdg_{1464} = \frac{- pdg_{1939} - \sqrt{pdg_{1939}^{2} - 4 pdg_{4231} pdg_{9139}}}{2 pdg_{9139}}\)
LHS diff is 0 RHS diff is (-pdg9139*sqrt((pdg1939**2 - 4*pdg4231*pdg9139)/pdg9139**2) + sqrt(pdg1939**2 - 4*pdg4231*pdg9139))/(2*pdg9139) 5982958248:
9999999968:
5982958248:
9999999968:
11.5 simplify
  1. 9582958293; locally 4433112:
    \(x = \sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a))\)
    \(pdg_{1464} = - \frac{pdg_{1939}}{2 pdg_{9139}} + \sqrt{\frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} - \frac{pdg_{4231}}{pdg_{9139}}}\)
  1. 9999999969; locally 8761200:
    \(x = \frac{-b+\sqrt{b^2-4ac}}{2 a}\)
    \(pdg_{1464} = \frac{- pdg_{1939} + \sqrt{pdg_{1939}^{2} - 4 pdg_{4231} pdg_{9139}}}{2 pdg_{9139}}\)
LHS diff is 0 RHS diff is (pdg9139*sqrt((pdg1939**2 - 4*pdg4231*pdg9139)/pdg9139**2) - sqrt(pdg1939**2 - 4*pdg4231*pdg9139))/(2*pdg9139) 9582958293:
9999999969:
9582958293:
9999999969:
5 add X to both sides
  1. 5938459282; locally 1212129:
    \(x^2 + (b/a)x = -c/a\)
    \(pdg_{1464}^{2} + \frac{pdg_{1464} pdg_{1939}}{pdg_{9139}} = - \frac{pdg_{4231}}{pdg_{9139}}\)
  1. 0004307451:
    \((b/(2 a))^2\)
    \(\frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}}\)
  1. 5928292841; locally 1120000:
    \(x^2 + (b/a)x + (b/(2 a))^2 = -c/a + (b/(2 a))^2\)
    \(pdg_{1464}^{2} + \frac{pdg_{1464} pdg_{1939}}{pdg_{9139}} + \frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} = \frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} - \frac{pdg_{4231}}{pdg_{9139}}\)
valid 5938459282:
5928292841:
5938459282:
5928292841:
4 subtract X from both sides
  1. 5958392859; locally 7777621:
    \(x^2 + (b/a)x+(c/a) = 0\)
    \(pdg_{1464}^{2} + pdg_{1464} + \frac{pdg_{4231}}{pdg_{9139}} = 0\)
  1. 0006644853:
    \(c/a\)
    \(\frac{pdg_{4231}}{pdg_{9139}}\)
  1. 5938459282; locally 1212129:
    \(x^2 + (b/a)x = -c/a\)
    \(pdg_{1464}^{2} + \frac{pdg_{1464} pdg_{1939}}{pdg_{9139}} = - \frac{pdg_{4231}}{pdg_{9139}}\)
LHS diff is pdg1464*(-pdg1939 + pdg9139)/pdg9139 RHS diff is 0 5958392859:
5938459282:
5958392859:
5938459282:
9 square root both sides
  1. 9385938295; locally 2985412:
    \((x+(b/(2 a)))^2 = -(c/a) + (b/(2 a))^2\)
    \(\left(pdg_{1464} + \frac{pdg_{1939}}{2 pdg_{9139}}\right)^{2} = \frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} - \frac{pdg_{4231}}{pdg_{9139}}\)
  1. 5982958249; locally 6608123:
    \(x+(b/(2 a)) = -\sqrt{(b/(2 a))^2 - (c/a)}\)
    \(pdg_{1464} + \frac{pdg_{1939}}{2 pdg_{9139}} = - \sqrt{\frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} - \frac{pdg_{4231}}{pdg_{9139}}}\)
  2. 9582958294; locally 6608102:
    \(x+(b/(2 a)) = \sqrt{(b/(2 a))^2 - (c/a)}\)
    \(pdg_{1464} + \frac{pdg_{1939}}{2 pdg_{9139}} = \sqrt{\frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} - \frac{pdg_{4231}}{pdg_{9139}}}\)
no check performed 9385938295:
5982958249:
9582958294:
9385938295:
5982958249:
9582958294:
8 LHS of expr 1 equals LHS of expr 2
  1. 5928292841; locally 1120000:
    \(x^2 + (b/a)x + (b/(2 a))^2 = -c/a + (b/(2 a))^2\)
    \(pdg_{1464}^{2} + \frac{pdg_{1464} pdg_{1939}}{pdg_{9139}} + \frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} = \frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} - \frac{pdg_{4231}}{pdg_{9139}}\)
  2. 5959282914; locally 1734000:
    \(x^2 + x(b/a) + (b/(2 a))^2 = (x+(b/(2 a)))^2\)
    \(pdg_{1464}^{2} + \frac{pdg_{1464} pdg_{1939}}{pdg_{9139}} + \frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} = \left(pdg_{1464} + \frac{pdg_{1939}}{2 pdg_{9139}}\right)^{2}\)
  1. 9385938295; locally 2985412:
    \((x+(b/(2 a)))^2 = -(c/a) + (b/(2 a))^2\)
    \(\left(pdg_{1464} + \frac{pdg_{1939}}{2 pdg_{9139}}\right)^{2} = \frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} - \frac{pdg_{4231}}{pdg_{9139}}\)
input diff is 0 diff is (pdg1464**2*pdg9139 + pdg1464*pdg1939 + pdg4231)/pdg9139 diff is -(pdg1464**2*pdg9139 + pdg1464*pdg1939 + pdg4231)/pdg9139 5928292841:
5959282914:
9385938295:
5928292841:
5959282914:
9385938295:
7 simplify
  1. 5928285821; locally 1239010:
    \(x^2 + 2 x (b/(2 a)) + (b/(2 a))^2 = (x + (b/(2 a)))^2\)
    \(pdg_{1464}^{2} + \frac{pdg_{1464} pdg_{1939}}{pdg_{9139}} + \frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} = \left(pdg_{1464} + \frac{pdg_{1939}}{2 pdg_{9139}}\right)^{2}\)
  1. 5959282914; locally 1734000:
    \(x^2 + x(b/a) + (b/(2 a))^2 = (x+(b/(2 a)))^2\)
    \(pdg_{1464}^{2} + \frac{pdg_{1464} pdg_{1939}}{pdg_{9139}} + \frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} = \left(pdg_{1464} + \frac{pdg_{1939}}{2 pdg_{9139}}\right)^{2}\)
valid 5928285821:
5959282914:
5928285821:
5959282914:
6 change variable X to Y
  1. 8582954722; locally 9091270:
    \(x^2 + 2 x h + h^2 = (x + h)^2\)
    \(pdg_{1464}^{2} + 2 pdg_{1464} pdg_{3410} + pdg_{3410}^{2} = \left(pdg_{1464} + pdg_{3410}\right)^{2}\)
  1. 0004858592:
    \(h\)
    \(pdg_{3410}\)
  2. 0000999900:
    \(b/(2 a)\)
    \(\frac{pdg_{1939}}{2 pdg_{9139}}\)
  1. 5928285821; locally 1239010:
    \(x^2 + 2 x (b/(2 a)) + (b/(2 a))^2 = (x + (b/(2 a)))^2\)
    \(pdg_{1464}^{2} + \frac{pdg_{1464} pdg_{1939}}{pdg_{9139}} + \frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} = \left(pdg_{1464} + \frac{pdg_{1939}}{2 pdg_{9139}}\right)^{2}\)
valid 8582954722: dimensions are consistent
5928285821:
8582954722: N/A
5928285821:
14 declare final expr
  1. 9999999968; locally 8811221:
    \(x = \frac{-b-\sqrt{b^2-4ac}}{2 a}\)
    \(pdg_{1464} = \frac{- pdg_{1939} - \sqrt{pdg_{1939}^{2} - 4 pdg_{4231} pdg_{9139}}}{2 pdg_{9139}}\)
no validation is available for declarations 9999999968:
9999999968:
15 declare final expr
  1. 9999999969; locally 8761200:
    \(x = \frac{-b+\sqrt{b^2-4ac}}{2 a}\)
    \(pdg_{1464} = \frac{- pdg_{1939} + \sqrt{pdg_{1939}^{2} - 4 pdg_{4231} pdg_{9139}}}{2 pdg_{9139}}\)
no validation is available for declarations 9999999969:
9999999969:
3 divide both sides by
  1. 9285928292; locally 8882098:
    \(ax^2 + bx + c = 0\)
    \(pdg_{1464}^{2} pdg_{9139} + pdg_{1464} pdg_{1939} + pdg_{4231} = 0\)
  1. 0002424922:
    \(a\)
    \(pdg_{9139}\)
  1. 5958392859; locally 7777621:
    \(x^2 + (b/a)x+(c/a) = 0\)
    \(pdg_{1464}^{2} + pdg_{1464} + \frac{pdg_{4231}}{pdg_{9139}} = 0\)
LHS diff is pdg1464*(pdg1939 - pdg9139)/pdg9139 RHS diff is 0 9285928292:
5958392859:
9285928292:
5958392859:
1 declare initial expr
  1. 9285928292; locally 8882098:
    \(ax^2 + bx + c = 0\)
    \(pdg_{1464}^{2} pdg_{9139} + pdg_{1464} pdg_{1939} + pdg_{4231} = 0\)
no validation is available for declarations 9285928292:
9285928292:
7.5 declare initial expr
  1. 8582954722; locally 9091270:
    \(x^2 + 2 x h + h^2 = (x + h)^2\)
    \(pdg_{1464}^{2} + 2 pdg_{1464} pdg_{3410} + pdg_{3410}^{2} = \left(pdg_{1464} + pdg_{3410}\right)^{2}\)
no validation is available for declarations 8582954722: dimensions are consistent
 8582954722: N/A
Physics Derivation Graph: Steps for quadratic equation derivation

Symbols for this derivation

See also all 227 symbols
symbol ID category latex scope dimension name value Used in derivations references
9139 variable a
\(a\)		 ['real'] dimensionless 45
1939 variable b
\(b\)		 ['real'] dimensionless 20
1464 variable x
\(x\)		 ['real'] dimensionless 140
3410 variable h
\(h\)		 real dimensionless none
  • str_note
2
4231 variable c
\(c\)		 ['real'] dimensionless 12
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