Solve from quadratic equations

1. abc-formula
2. pq-formula

abc-formula

ax2 + bx + c	= 0
x2 + b/a·x + c/a	= 0
x2 + b/a·x	= -c/a
x² + b/a·x + (b/(2a))²	= (b/(2a))² - c/a
(x + b/(2a))²	= (b/(2a))² - c/a
x + b/(2a)	= ± √((b/(2a))² - c/a) | - b/(2a)
x1,2	= -b/(2a) ± √((b/(2a))² - c/a)
	= -b/(2a) ± √(b² - 4ac)/(2a)
	= (-b ± √(b² - 4ac))/(2a)

This formula is also called ABC formula or “midnight formula”.

pq-formula

ax² + bx + c = 0 | :a

⇒ 0 = x² + px + q

The solution formula is:

x2 + px + q	= 0
x2 + px	= -q
x2 + px + (p/2)2	= (p/2)2 - q
(x + p/2)2	= (p/2)2 - q
x + p/2	= ± √((p/2)2 - q) | - p/2
x1,2	= -p/2 ± √((p/2)2 - q)

Both systems supply the same solutions.

Example:

0=2x²-4x-16
and so: a=2, b=4, c=-16

OR:

⇒0=x²-2x-8
and so: p=2, q=-8

x₁=4
x₂=-2

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