derivation of quadratic formula
Since A is nonzero, we can divide by A and obtain the equation
x2+bx+c=0, |
where b=BA and c=CA. This equation can be written as
x2+bx+b24-b24+c=0, |
so completing the square, i.e., applying the identity (p+q)2=p2+2pq+q2, yields
(x+b2)2=b24-c. |
Then, taking the square root of both sides, and solving for x, we obtain
the solution formula
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
ax2+bx+c=0 |
by 4a, resulting in the equation
4a2x2+4abx+b2=b2-4ac, |
in which the left-hand side can be expressed as (2ax+b)2. From here, the proof is identical.
Title | derivation of quadratic formula |
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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 |