A translation service to convert from standard form, ax²+bx+c, to the factored, d(ex+h)(fx+g), and vertex, a(x-x₀)²+y₀, forms.
dotnet run a=... b=... c=...
d(ex+h)(fx+g)
= d(efx² + (eg+fh)x + gh)
a = d * ef
b = d * (eg + fh)
c = d * gh
d = gcd(a, b, c)
a' = a / d = ef
b' = b / d = eg + fh
c' = c / d = gh
To perform the translation from a, b, & c requires factoring to get to necessary values of e, f, g, & h.
ax²+bx
d = gcd(a, b)
a' = a / d
b' = b / d
d(a'x² + b'x)
= dx(a'x + b')
= d(x + 0)(a'x + b')
e = 1
h = 0
f = a'
g = b'
a(x-x₀)²+y₀
= a(x² - 2x₀x + x₀²) + y₀
= ax² - 2ax₀x + ax₀² + y₀
a = a
b = -2ax₀
c = ax₀² + y₀
x₀ = -b / 2a
y₀ = c - ax₀²
= c - a(-b / 2a)²
= c - b² / 4a