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# difference between r and r square

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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\,\!.

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:

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?