geno3141

Factor  (x² + y² - z²)² - 4x²y²

--->      (x² + y² - z²)² - (2xy)²

Factor as the difference of two squares:

--->     [ ( x² + y² - z² ) + (2xy) ] · [ ( x² + y² - z² ) - (2xy) ] 

Rewrite:

--->     [ x² + 2xy + y² - z² ] · [  x² - 2xy + y² - z² ] 

Factor:

--->     [ (x + y)² - z² ] · [  (x - y)² - z² ] 

Factor as the difference of two squares:

--->     [ (x + y + z)(x + y - z) ] · [ (x - y + z)(z - y - z) ]

Answer:   (x + y + z)(x + y - z)(x - y + z)(z - y - z) 

I don't know how to do these algebraicly, so I graphed them. These are the answers that I got:

a  =  2.599364238

b  =  1.226651563

c  =  2.370956068

d  =  1.455059732

Their product is 11.

Since triangle(ACB) is a right triangle with base AC, point E will be the midpoint of the base.

Look at right triangle(AEB). The hypotenuse is AB = 5; one side is AE = 3, so the other side, BE = 4 (Pythagorean Theorem).

Therefore, the area of triangle(ABC) = ½·AC·BE  =  ½·6·4  =  12  and the area of triangle(BEC)  =  6.

Now I want to find the length of AD.

AD is the height of triangle(ABC) if BC is the base.

Area of triangle(ABC)  =  12  =  ½·BC·AD     --->     12  =  ½·5·AD     --->     12  =  2.5·AD     --->     

AD  =  4.8

I will use the length of AD to find the length of BD.  

Triangle(ABD) is a right triangle with hypotenuse AB = 5 and side AD = 4.8. 

Therefore, using the Pythagorean Theorem, BD  = 1.4.

Since BC  =  5  and  BD  =  4,  CD  =  3.6.

If you look at triangle(BCE) with base BC, triangle(DCE) with base DC, and tiangle(BDE) with base BD, they all have the same height.  (It isn't drawn, but you can draw it from point E perpendicular to BC.

Area of triangle(BCE)  =  area of triangle(DCE)  + area of triangle(BDE)

Since the length of DC = 3.6 and the length of BD = 1.4 and length of CD = 5, the length of CD = 3.6/5 the length of BC.

So, the area of triangle(DCE)  =  3.6/5 · area of triangle(BCE)

--->       area of triangle(DCE)  =   3.6/5 · 6  =  4.32                   

A quadratic equation can have exactly one solution if it is a perfect square.

(2x + 7)(x - 5)  =  -43 + jx

2x² - 10x + 7x - 35  =  -43 + jx                                   [multiply out]

 2x² - 3x - 35  =  -43 + jx                                            [simplify]

2x² - 3x - jx - 35 + 43  =  0                                         [subtract -43 + jx from both sides]

 2x² - (3 + j)x + 8  =  0                                               [factor and simplify]

2[ x² - ( (3 + j)/2 )x + 4 ]  =  0                                     [factor out the 2]

x² - ( (3 + j)/2 )x + 4   =  0                                          [divide both sides by 2]

Now:  x² - 4x + 4  =  (x - 2)²  =  0     and     x² + 4x + 4  =  (x + 2)²​  =  0      will have one real solution each.

So:  either  (3 + j)/2   =  4     or      (3 + j)/2   =  -4

Solving:       3 + j  =  8          or         3 + j  =  -8

                          j  =  5         or               j  =  -11

Look at triangle(ABC)  --  Drop a perpendicular from C to AB.

This will be the height of triangle(CQB) with base QB.

It will also be the height of triangle(CQA) with base AQ.

Since AQ = 3 x QB, the area of triangle(CQA) = 3 x area of triangle(CQB).

Look at triangleAQC)  --  Drop a perpendicular from Q to AC.

This will be the height of triangle(QRA) with base RA.

It will also be the height of triangle(QRC) with base RC.

Since AR/RC = 2/3, the area of triangle(QRA) = 2/3 · area of triangle(QRC).

This means that triangle(QRC) = 3/5 the area of triangle(CQA).

Can you take it from here?

There are 4 ways to select a science textbook to in the middle. (There are now 3 science books remaining.)

There are 5 ways to select a history textbook to place a on the left-hand end. (There are now 4 history books remaining.)

There are 4 ways to select a history textbook to place on the right-hand end. (There are now 3 history books remaining.)

With 6 books remaining, you can fill the other 6 spaces in 6 x 5 x 4 x 3 x 2 x 1 ways.

The total number of ways is:  4 x 5 x 4 x 6 x 5 x 4 x 3 x 2 x 1 ways.

Since it's symmetrical, there are two answers:  q = 3 and p = 3/2  or  p = 3 and q = 3/2

Draw a rectangle.

I believe that these form the "pentalope numbers":  1, 5, 15, 35, 70, 126, 210, ...

Since there won't be any line segments that connect non-adjacent points until you have at least 4 points:

4 points will result in 1 point of intersection,

5 points will result in 5 points of intersection,

6 points will result in 15 points of intersection,

...

10 points will result in 210 points of intersection.

The formula is:  [n · (n + 1) · (n + 2) · (n + 3) ] / 24

You can take these two factors:  (-x - y + z)(-x + y + z)

and factor -1 out of these:          [-1(x + y - z)]·[-1(x - y - z)]

Rewrite:                                     (-1)(-1)·(x + y - z)(x - y - z)

to get:                                               (x + y - z)(x - y - z)