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derivation of quadratic formula

(Proof)

Suppose are real numbers, with $A\neq 0$ , and suppose

$\displaystyle Ax^2+Bx+C=0.$

Since is nonzero, we can divide by and obtain the equation

$\displaystyle x^2 + bx+c = 0,$

where $b=\frac{B}{A}$ and $c=\frac{C}{A}$ . This equation can be written as

$\displaystyle x^2 + bx + \frac{b^2}{4} -\frac{b^2}{4} +c =0,$

, yields

$\displaystyle \left(x+\frac{b}{2}\right)^2 = \frac{b^2}{4} - c.$

Then, taking the square root of both sides, and solving for

, we obtain the solution formula

$\displaystyle x$	$\displaystyle =$	$\displaystyle -\frac{b}{2} \pm \sqrt{\frac{b^2}{4}-c}$
	$\displaystyle =$	$\displaystyle \frac{B}{2A} \pm \sqrt{\frac{B^2}{4A^2}-\frac{C}{A}}$
	$\displaystyle =$	$\displaystyle \frac{-B\pm\sqrt{B^2-4AC}}{2A},$

and the derivation is completed.

A slightly less intuitive but more aesthetically pleasing approach to this derivation can be achieved by multiplying both sides of the equation

$\displaystyle ax^2+bx+c=0$

, resulting in the equation

$\displaystyle 4a^2x^2+4abx+b^2=b^2-4ac,$

in which the left-hand side can be expressed as

. From here, the proof is identical.

"derivation of quadratic formula" is owned by mathcam. [ full author list (3) | owner history (2) ]

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See Also: quadratic formula, quadratic equation in $\mathbb{C}$

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Cross-references: proof, derivation, solution, sides, square root, identity, completing the square, equation, divide, real numbers
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This is version 8 of derivation of quadratic formula, born on 2001-11-07, modified 2006-07-22.
Object id is 700, canonical name is DerivationOfQuadraticFormula.
Accessed 19758 times total.

Classification:

12D10 (Field theory and polynomials :: Real and complex fields :: Polynomials: location of zeros )

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