PlanetMath: strange root

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (211)
Orphanage
Unclass'd (1)
Unproven (473)
Corrections (38)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

strange root

(Definition)

In solving certain types of equations, one may obtain besides the proper (right) roots also some strange roots which do not satisfy the original equation. Such a thing can happen especially when one has in some stage squared both sides of the treated equation; in this situation one must check all “roots” by substituting them to the original equation.

Example.

$\displaystyle x-\sqrt{x} = 12$

$\displaystyle x-12 = \sqrt{x}$

$\displaystyle (x-12)^2 = (\sqrt{x})^2$

$\displaystyle x^2-24x+144 = x$

$\displaystyle x^2-25x+144 = 0$

$\displaystyle x = \frac{25\pm\sqrt{25^2-4\cdot144}}{2} = \frac{25\pm7}{2}$

$\displaystyle x = 16 \quad \lor \quad x = 9$

Substituting these values of

into the left side of the original equation yields

$\displaystyle 16-4 = 12, \quad 9-3 = 6.$

Thus, only

is valid,

is a strange root. (How

is related to the solved equation, is explained by that it may be written $(\sqrt{x})^2-\sqrt{x}-12 = 0$ , from which one would obtain via the quadratic formula that $\sqrt{x} = \frac{1\pm7}{2}$ , i.e. $\sqrt{x} = 4$ or $\sqrt{x} = -3$ . The latter corresponds the value

, but it were relevant to the original equation only if we would allow negative values for square roots of positive numbers; the current practice excludes them.)

The general explanation of strange roots when squaring an equation is, that the two equations

$\displaystyle a = b,$

$\displaystyle a^2 = b^2$

are not equivalent (but the equations $a = \pm b$ and

would be such ones).

"strange root" is owned by pahio.

(view preamble)

Other names:

wrong root

This object's parent.

Log in to rate this entry.
(view current ratings)

Cross-references: positive, square roots, negative, quadratic formula, sides, roots, equations
There is 1 reference to this entry.

This is version 4 of strange root, born on 2008-03-20, modified 2008-03-21.
Object id is 10426, canonical name is StrangeRoot.
Accessed 221 times total.

Classification:

AMS MSC:	26A09 (Real functions :: Functions of one variable :: Elementary functions)
	97D99 (Mathematics education :: Education and instruction in mathematics :: Miscellaneous)

Pending Errata and Addenda

None.[ View all 1 ]

Discussion

forum policy

No messages.

Interact