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From Wikipedia

Difference of two squares

In mathematics, the difference of two squares, or the difference of perfect squares, is when a number is squared, or multiplied by itself, and is then subtracted from another squared number. It refers to the identity

a^2-b^2 = \left(a-b\right)\left(a+b\right)

from elementary algebra.

Proof

The proof is straightforward, starting from the RHS: apply the distributive law to get a sum of four terms, and set as an application of the commutative law. The resulting identity is one of the most commonly used in all of mathematics.

ba - ab = 0\,\!

The proof just given indicates the scope of the identity in abstract algebra: it will hold in any commutative ringR.

Also, conversely, if this identity holds in a ringR for all pairs of elements a and b of the ring, then R is commutative. To see this, we apply the distributive law to the right-hand side of the original equation and get

a^2 - ab + ba - b^2\,\!

and if this is equal to a^2 - b^2, then we have

a^2 - ab + ba - b^2 - \left(a^2 - b^2\right) = 0\,\!

and by associativity and the rule that r-r=0, we can rewrite this as

ba - ab = 0.\,\!

If the original identity holds, then, we have ba - ab = 0 for all pairs a, b of elements of R, so the ring R is commutative.

In geometry

The difference of two squares can also be illustrated geometrically as the difference of two square areas in a plane. In the diagram, the shaded part represents the difference between the areas of the two squares, i.e. a^2 - b^2\,\!. The area of the shaded part can be found by adding the areas of the two rectangles; a(a-b) + b(a-b)\,\!, which can be factorized to (a+b)(a-b)\,\!. Therefore a^2 - b^2 = (a+b)(a-b)\,\!

Uses

Complex number case: sum of two squares

The difference of two squares is used to find the linear factors of the sum of two squares, using complex number coefficients.

For example, the root of z^2 + 5\,\! can be found using difference of two squares:

z^2 + 5\,\!
= z^2 - (\sqrt{-5})^2
= z^2 - (i\sqrt5)^2
= (z + i\sqrt5)(z - i\sqrt5)

Therefore the linear factors are (z + i\sqrt5) and (z - i\sqrt5).

Rationalising denominators

The difference of two squares can also be used in the rationalising of irrationaldenominators. This is a method for removing surds from expressions (or at least moving them), applying to division by some combinations involving square roots.

For example: The denominator of \dfrac{5}{\sqrt{3} + 4}\,\! can be rationalised as follows:

\dfrac{5}{\sqrt{3} + 4}\,\!
= \dfrac{5}{\sqrt{3} + 4} \times \dfrac{\sqrt{3} - 4}{\sqrt{3} - 4}\,\!
= \dfrac{5(\sqrt{3} - 4)}{(\sqrt{3} + 4)(\sqrt{3} - 4)}\,\!
= \dfrac{5(\sqrt{3} - 4)}{\sqrt{3}^2 - 4^2}\,\!
= \dfrac{5(\sqrt{3} - 4)}{3 - 16}\,\!
= -\dfrac{5(\sqrt{3} - 4)}{13}\,\!

Here, the irrational denominator \sqrt{3} + 4\,\! has been rationalised to 13\,\!.



From Yahoo Answers

Question:Excel tells me that r-squared is 0.851 so what is the difference between r-squared and r-squared (adjusted) and how do I find it. If you need the points they are : (0,100) (0.5,105) (1,110) (2,120) (1.5,120) (2.5,180) (3,200) (4.5,220) (6,250) (9,255).

Answers:r-squared tells you how much of the variability observed in your data is accounted for by the model. The adjusted r-squared modifies r-squared by taking into account the number of covariates or predictors you include in your model. See the link below for details.

Question:

Answers:R2 is a statistic that will give some information about the goodness of fit of a model. In regression, the R2 coefficient of determination is a statistical measure of how well the regression line approximates the real data points. An R2 of 1.0 indicates that the regression line perfectly fits the data. Adjusted R2 is a modification of R2 that adjusts for the number of explanatory terms in a model. Unlike R2, the adjusted R2 increases only if the new term improves the model more than would be expected by chance. The adjusted R2 can be negative, and will always be less than or equal to R2. Adjusted R2 does not have the same interpretation as R2. As such, care must be taken in interpreting and reporting this statistic. Adjusted R2 is particularly useful in the Feature selection stage of model building. Adjusted R2 is not always better than R2: adjusted R2 will be more useful only if the R2 is calculated based on a sample, not the entire population. For example, if our unit of analysis is a state, and we have data for all counties, then adjusted R2 will not yield any more useful information than R2.

Question:Doing some maths revision and i got a bit confused? Do they both calculate area of a circle? Or are they different? Thanks in advance

Answers:No they're different. 2 Pi r is for calculating the circumference of a circle while Pi r 2 is for calculating the area of a circle.

Question:i bought a pair of hollister jeans and i want to fit into them bye september first, or eventually. and the value village tag said size 3-4??? help?

Answers:1R = 1 regular as opposed to 1 short or 1 tall is that what you were looking for?

From Youtube

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LM-R : diginfonews :The Ferrari F1 racecar set a record winning double titles. Crucial to this achievement was the incorporation of aluminum forged 2 piece wheels featuring the world-renowned BBS seven cross-spoke design. There are two general classifications in the manufacture of light alloy metal wheels: casting and forging. The forging of Japanese swords has a long and detailed history. Forging is a process that uses heat and pressure to fashion metal into a desired shape. In the forging process of BBS wheels, a billet is heated to around 450 C while a pressure of more than 4 tons per square centimeter is applied to cast the alloy between forging dies. In contrast to casting, forging is not suited for mass-production of aluminum wheels. BBS has selected this technologically complex forging process over casting because it enables the company to produce lightweight, high-performance wheels. In the casting process, when the molten metal is poured into the cast die, air mixes in with the molten metal, and develops tiny air pockets as the metal cools and hardens, resulting in a grainy metallic structure. On the other hand, with forging there are absolutely no air pockets, and this in turn results in a compact high integrity metallic structure and enhances the strength of the forged metal. For these reasons then, BBS has selected this technologically complex forging process because it enables the company to produce lightweight, high-performance wheels.