derivation of quadratic formula

Suppose A,B,C are real numbers, with A0, and suppose


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


where b=BA and c=CA. This equation can be written as


so completing the square, i.e., applying the identity (p+q)2=p2+2pq+q2, yields


Then, taking the square root of both sides, and solving for x, we obtain the solution formulaMathworldPlanetmathPlanetmath

x = -b2±b24-c
= B2A±B24A2-CA
= -B±B2-4AC2A,

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


by 4a, resulting in the equation


in which the left-hand side can be expressed as (2ax+b)2. From here, the proof is identical.

Title derivation of quadratic formula
Canonical name DerivationOfQuadraticFormula
Date of creation 2013-03-22 11:56:44
Last modified on 2013-03-22 11:56:44
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 12
Author mathcam (2727)
Entry type Proof
Classification msc 12D10
Related topic QuadraticFormula
Related topic QuadraticEquationInMathbbC