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why does the quadratic formula always work
It always works, with one catch we'll meet at the end, because it's completing a square. Try x squared plus six x equals seven: a square, plus two strips three wide.
Fill the corner, three by three, nine, and add nine to both sides. Now it's a perfect square: x plus three squared is sixteen, so x plus three is four or minus four.
Now do it for any quadratic. Divide by a. Then add b over two a, squared, to both sides, the missing corner, so the left side becomes a perfect square.
Take the square root, remembering plus or minus, and move b over two a across. Out falls the formula. Every step is plain algebra that can be undone, so it always works.
One catch hides in the root. Under it sits b squared minus four a c. Positive: the parabola crosses the axis twice. Zero: it touches once. Negative: it floats clear, no real answers.
So yes, it always works, for any quadratic with a not zero. Real answers when that discriminant isn't negative, complex ones when it is. Same square, completed once.
The formula is one completed square, worked out once so you never have to do it again.