CPhill

LOL!!!!

Maybe we have a "Conjunctive Contraction"   ....[ or is it....a "Contractive Conjunction ?? " ]

The fallacy is the transition from Step 4 to Step 5

4    (a + b) ( a - b)   = b ( a - b)

But if   a   = b.....then we have

5   (a + b) ( 0)  = b ( 0) 

So....6   becomes

0    =    0         which is the logical conclusion, not    " a + a   =   a "

Thanks, EP.....sometimes....my brain actually works!!!

Mmmmmm....not sure about all perfect squares , GA

9    =   2 + 3 + 4

25 = 3 + 4 + 5 + 6 + 7

36 = 11 + 12 + 13

81 = 26 + 27 + 28

etc.....

An altitude drawn to the hypotenuse of an isosceles right triangle always bisects the hypotenuse

So   BD = CD

And  AD forms a geometric mean of  BD and CD

So....we have this relationship

BD / AD   = AD / CD         but BD = CD  ...so...

BD / 5 = 5 / BD

BD^2   = 25

BD    =  5

But BD is 1/2 of base BC

And AD is an altitude of ABC drawn to this base

So....the area of ABC   = (1/2)BC * AD   =  5 * 5    =   25 units^2

Also

4) 4(2-a)^2 -81     is a difference of squares

[ 2 (2-a) + 9 ]  [ 2(2-a) - 9 ]   =

[4 - 2a + 9 ]  [ 4 - 2a - 9 ]    =

[13 - 2a ] [ -5 - 2a ] =       { factor a "-1" from the last term....move it to the front }

- [13 - 2a ] [ 5 + 2a ]

[ 2a - 13 ] [ 2a + 5 ]

We can actually factor 3 as a difference of squares  if we note that

x  = √x * √x

16 - 25 x   =

(4 + 5√x ) ( 4 - 5√x )

We can use the Law  of Sines to solve this

Since B is the right angle   and the triangle is isosceles....then  angles  A and C = 45°

The incenter of the triangle is the point where the three angle bisectors meet

One of the triangles formed by their intersection is ΔBAL

Since B is bisected....then angle ABL    =  45°

And since A is bisected....angle BAL   = 22.5°

And angle ALB   =  180 - 45 - 22.5   = 112.5°

So

BL                       AB

_______   =      _______

sin BAL            sin ALB

BL                    4sqrt(2)

____    =        _________

sin 22.5           sin 112.5

                     4sqrt (2) sin 22.5

BL =            _______________     ≈    2.34

                          sin 112.5

√[ 48x^4y^3 ]   = √16 *√3 * √x^4 *√y^3   =  4 * √3 * x^2  *y√y  =  

4x^2y√ [ 3y]

Second one.....easiest way to do this is to  simplify top/bottom separately

1     +      2              [ y   + 2x^2]

__          __     =        ____________     (1)        and

x^2          y                    x^2y

5     -   6                   [ 5y^2 - 6x ]

__      ____    =      ___________        (2)

x         y^2                   xy^2

So we have    (1)   / (2)

Keep (1)   , invert (2)   and multiply

[ y + 2x^2]         [xy^2]

_________   *  _________     =

 x^2y                [ 5y^2 - 6x]

[ y + 2x^2]        [ xy^2 ]

________    *  _______ =

 [5y^2 - 6x ]       [x^2y]

[ y + 2x^2 ] *  [  y ]

_________   _____    =

[ 5y^2 - 6x]     [ x]

[ y^2 + 2x^2y ]

____________

[ 5xy^2 - 6x^2]