## Actual timed Mathomatic output from the quadratic script

```Mathomatic version 16.0.2
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−>
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−> pause
2−> ; 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 equation
3−> simplify

x·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−> pause
3−> 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−> pause
3−> x
level 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−> pause
3−> set no debug
Success.
3−> repeat simplify

1
((((b^2 − (4·c·a))^–)·sign0) − b)
2
#3: x = –––––––––––––––––––––––––––––––––
(2·a)

3−> compare with 2
Comparing #2 with #3...
Equations are identical.
3−>