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Therefore, the solution = is extraneous and not valid, and the original equation has no solution. For this specific example, it could be recognized that (for the value x = − 2 {\displaystyle x=-2} ), the operation of multiplying by ( x − 2 ) ( x + 2 ) {\displaystyle (x-2)(x+2)} would be a multiplication by zero.
These paradoxes may be due to fallacious reasoning , or an unintuitive solution . The term paradox is often used to describe a counter-intuitive result. However, some of these paradoxes qualify to fit into the mainstream viewpoint of a paradox, which is a self-contradictory result gained even while properly applying accepted ways of reasoning .
[1] [2] This iconic portrait is an example of a photograph that is very well known by the general public as a real photograph and not an altered one. Another is exampled in the Library of Congress Prints and Photographs Online Catalogue wherein it exposes a manipulated American Civil War photograph of General Ulysses S. Grant posing horseback ...
In other words, a real image is an image which is located in the plane of convergence for the light rays that originate from a given object. Examples of real images include the image produced on a detector in the rear of a camera, and the image produced on an eyeball retina (the camera and eye focus light through an internal convex lens).
It follows that the solutions of E are a subset of those of E'. The solution set of E' consists of the solution set of E plus, possibly, some other solutions, called extraneous solutions. --Lambiam 15:43, 19 January 2008 (UTC) Lambiam, on your first example, sorry, why isn't 2 a solution? Tparameter 16:08, 19 January 2008 (UTC)
An example of a nonlinear delay differential equation; applications in number theory, distribution of primes, and control theory [5] [6] [7] Chrystal's equation: 1 + + + = Generalization of Clairaut's equation with a singular solution [8] Clairaut's equation: 1
The worked-example effect is a learning effect predicted by cognitive load theory. [1] [full citation needed] Specifically, it refers to improved learning observed when worked examples are used as part of instruction, compared to other instructional techniques such as problem-solving [2] [page needed] and discovery learning.
To illustrate, the solution + = has bases with a common factor of 3, the solution + = has bases with a common factor of 7, and + = + has bases with a common factor of 2. Indeed the equation has infinitely many solutions where the bases share a common factor, including generalizations of the above three examples, respectively