## review derivation: quadratic equation derivation

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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

symbol ID category latex scope dimension name value Used in derivations references
3410 variable h
$$h$$
real
none
• str_note
2
1939 variable b
$$b$$
['real']
20
4231 variable c
$$c$$
['real']
12
9139 variable a
$$a$$
['real']
45
1464 variable x
$$x$$
['real']
140
MESSAGE:
• saved to file