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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.


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)\,\!


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:A) A square must have 4 right angles, but a rhombus may or may not have 4 right angles. B) A rhombus has 4 right angles, and a square does not. C) A rhombus is always bigger than a square. D) A square has sides that are parallel, but a rhombus does not.

Answers:The answer is A. A rhombus is a parallelogram with 4 equal sides, but not necessarily connected via right angles. For what it's worth, a square is technically a rhombus.

Question:--What is the difference between a trapezoid and an isosceles trapezoid? ---What kinds of angles are formed by the diagonals of a kite, square, rhombus?

Answers:Right angles. A trapezoid is a quadrilateral (four-sided polygon) that has two of the sides parallel. In an isosceles trapezoid this is still true, but the other two sides are of equal length, and the base angles are equal as well.

Question:Can anyone answer these? {1} Name a characteristic for a rhombus {2} A picture frame is shaped like a regular hexagon. One of the angles has a measure of 120 degrees. What is the sum of the remaining angle measures? {3} What is the distance around a regular heptagon if each side is 112 inches long? {4} Each side of a regular pentagon is 24 inches long. Find the perimeter of the pentagon.

Answers:1) A rhombus has four sides of equal length. It is a parallelogram, that is the first and third sides are parallel and the second and fourth sides are parallel. The only difference between a rhombus and a square is that the square must have 4 90 degree angle, while the rhombus has n degrees, 180 - n degrees, n degrees, and 180 - n degrees where n is any angle such that 0 < n < 180. -------------- 2) A regular hexagon has all six angles equal to 120 degrees. The other five angles add up to 5 * 120 = 600 degrees. -------------- 3) A heptagon has seven sides and a regular heptagon has seven equal sides. So the "distance around the heptagon" is 7 * 112 = 784 inches. -------------- 4) A pentagon has five sides and a regular pentagon has 5 equal sides. So the perimeter is: 5 * 24 = 120 inches. .

Question:draw a square. draw a rhombus that is not a square. compare the two figures. how are they alike?different?

Answers:A diamond with sides of equal length is a rhombus.

From Youtube

8.4 What's Similar and Different between Rhombuses, Rectangles, Squares? :8.4 What's Similar and Different between Rhombuses, Rectangles, Squares? We saw that rhombuses, rectangles, and squares are all parallelograms, but they're a bit more complex than parallelograms. Rhombus: 4 sides congruent, diagonals perpendicular, diagonals bisect opposite angles Rectangle: 4 angles congruent Square: 4 sides congruent and 4 angles congruent