Quadratic equation

In mathematics, a quadratic equation is a polynomial equation of the second degree. The generalized form is

Derivation

The quadratic formula is derived by the method of completing the square.

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:ax^2+bx+c=0

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Dividing our quadratic equation by a (which is allowed because a is non-zero), we have

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:

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x^2 + rac{b}{a} x + rac{c}{a}=0

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which is equivalent to

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:x^2+rac{b}{a}x=-rac{c}{a}.

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The equation is now in a form in which we can conveniently complete the square. To "complete the square" is to add a constant (i.e., in this case, a quantity that does not depend on x) to the expression to the left of "=", that will make it a perfect square trinomial of the form

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x2 + 2xy + y2. Since "2xy" in this case is (b/a)x, we must have y = b/(2a), so we add the square of b/(2a) to both sides, getting

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:x^2+rac{b}{a}x+rac{b^2}{4a^2}=-rac{c}{a}+rac{b^2}{4a^2}.

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The left side is now a perfect square; it is the square of (x + b/(2a)). The right side can be written as a single fraction; the common denominator is 4a2. We get

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:left(x+rac{b}{2a} ight)^2=rac{b^2-4ac}{4a^2}.

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Taking square roots of both sides yields

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:x+rac{b}{2a}=pmrac{sqrt{b^2-4ac }}{2a}.

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Subtracting b/(2a) from both sides, we get

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:x=rac{-b}{2a}pmrac{sqrt{b^2-4ac }}{2a}=rac{-bpmsqrt{b^2-4ac }}{2a}.

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