Mathomatic version 16.0.2 Copyright © 1987-2012 George Gesslein II. 200 equation spaces available in RAM; 2 megabytes per equation space. HTML color mode enabled; manage by typing "help color". Reading in file specified on the command line... 1−>real 0.00 user 0.00 sys 0.00 seconds total execution time of this script.1−>; General quadratic (2nd degree polynomial) formula.1−>; Formula for the 2 roots (solutions for x)1−>; of the general quadratic equation.1−>;1−>a x^2 + b x + c = 0 ; The general quadratic equation.#1: (a·x^2) + (b·x) + c = 0 1−>copy select ; Make a copy and select it.#2: (a·x^2) + (b·x) + c = 0 2−>solve verifiable for x ; Mathomatic can easily solve and verify that:Solving equation #2 for x with required verification... Equation is a degree 2 polynomial equation in x. Equation was solved with the quadratic formula. Solve and "simplify quick" successful: 1 ((((b^2 − (4·a·c))^–)·sign) − b) 2 #2: x = –––––––––––––––––––––––––––––––– (2·a) All solutions verified. 2−>; This is the quadratic formula.2−>; The coefficients (a, b, and c) may be any mathematical expression not containing x.2−>pause2−>; Here is the derivation and proof of the quadratic formula,2−>; without actually using the quadratic formula,2−>; because that is what we are trying to derive now, from the quadratic equation:2−>#1:#1: (a·x^2) + (b·x) + c = 0 1−>copy select ; make a copy of the general quadratic equation to work on and select it.#3: (a·x^2) + (b·x) + c = 0 3−>-=c ; subtract "c" from both sides.#3: (a·x^2) + (b·x) = -c 3−>/=a ; divide both sides by "a".((a·x^2) + (b·x)) -c #3: ––––––––––––––––– = –– a a 3−>pause Next simplify it and turn it into a repeated factor polynomial equation3−>simplifyx·b -c #3: x^2 + ––– = –– a a 3−>+=b^2/(4*(a^2)) ; add "b^2/(4*(a^2))" to both sides.x·b b^2 b^2 c #3: x^2 + ––– + ––––––– = ––––––– − – a (4·a^2) (4·a^2) a 3−>; Now the LHS is a repeated factor polynomial, next factor it by pressing Enter to simplify.3−>pause3−>simplify ; Now the LHS is a factored polynomial, so solving for the single "x" is easy.b b (((2·x) + –)^2) (–^2) a a c #3: ––––––––––––––– = ––––– − – 4 4 a 3−>set debug 1 ; Let Mathomatic do the work and show it too.Success. 3−>; Show how easy it is to solve this equation now, after pressing Enter.3−>pause3−>xlevel 1: 0.25*(((2*x) + (b/a))^2) = ((0.25*b^2) - (c*a))/a^2 Dividing both sides of the equation by "0.25": level 1: ((2*x) + (b/a))^2 = 4*((0.25*b^2) - (c*a))/a^2 Raising both sides of the equation to the power of 0.5: level 1: (2*x) + (b/a) = ((4*((0.25*b^2) - (c*a))/a^2)^0.5)*sign0 Subtracting "b/a" from both sides of the equation: level 1: 2*x = (((4*((0.25*b^2) - (c*a))/a^2)^0.5)*sign0) - (b/a) Dividing both sides of the equation by "2": level 1: x = 0.5*((((4*((0.25*b^2) - (c*a))/a^2)^0.5)*sign0) - (b/a)) Solve completed: level 1: x = 0.5*((((4*((0.25*b^2) - (c*a))/a^2)^0.5)*sign0) - (b/a)) Solve successful: b^2 4·(––– − (c·a)) 4 1 b (((–––––––––––––––^–)·sign0) − –) a^2 2 a #3: x = ––––––––––––––––––––––––––––––––– 2 3−>; Here is the raw solve result, press the Enter key to simplify and compare with the quadratic formula.3−>pause3−>set no debugSuccess. 3−>repeat simplify1 ((((b^2 − (4·c·a))^–)·sign0) − b) 2 #3: x = ––––––––––––––––––––––––––––––––– (2·a) 3−>compare with 2Comparing #2 with #3... Equations are identical. 3−>Successfully finished reading script file "quadratic.in". 3−> End of input.

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