The quadratic formula is probably the simplest way to solve anunfactorable quadratic. It is an equation in which you plug in the a term, b term,and c term, simplify and vola! You're done! Deriving an equation is taking a generalform of something and turning it into a useble equation. We do this by completingthe square.
Start with ax² + bx + c = 0
Bring the c over: ax² + bx = -c
Now use Completing the Square
Divide by a: x² + (b/a)x = -(c/a)
Use the Completing the Square formula: x² + (b/a)x + (b/2a)² = -(c/a) + (b/2a)²
Factor and expand the right side: (x + b/2a)² = -c/a + b²/4a²
Simplify the right side: (x + b/2a)² = (-4ca + b²)/4a²
Simplify again: (x + b/2a)² = (b² - 4ac)/4a²
Take the square route of both sides: (x + b/2a) = ± Ö(b² - 4ac)/2a
Move the b/2a over: x = -b/2a ± Ö(b² - 4ac)/2a
Simplify one last time and vola! There it is: x = (-b ± Ö(b²- 4ac))/2a
From now on, if know a, b, and c, you can easily figure out the equationby plugging them in to the nice quadratic formula and simplifying. See, easy as a,b, c.
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