CPhill

3n + 4001 + 3n + 4001 + 3n + 4001 =

 (3n + 4001) + (3n + 4001) + ( 3n + 4001) =

3 ( 3n + 4001)  =

9n + 12003

a^2 - 6ab - 7b^2

(a  - 7b) (a + b)

14e - 7f  =  28

 7e +   f =   50 

Note that if we divide the top equation through by 7, we have

2e - f = 4      add this to the second equation  and we get

9e = 54        divide both sides by 9

e = 54/9  =  6

And using the second equation to find f, we have

7(6) + f  = 50

42 + f  =  50       subtract 42 from both sides

f  =  8

y^2 +  2y  + 12x + 25 = 0     

Subtract y^2 + 2y + 25 from both sides

12x = -y^2 - 2y -25        divide both sides by 12

x = (-1/12)y^2 - (1/6)y -  25/12      this is the general form

y^2 +  2y  + 12x + 25 = 0

Now....let's find the vertex form

Subtract   12x, 5 from both sides

y^2 + 2y   =    -12x  -  25

Complete the square on y

Take (1/2) of 2  = 1....square this  = 1^2  = 1 and add to both sides

y^2 + 2y + 1  =  -12x  - 25 + 1

y^2 + 2y + 1  =  -12x  - 24     factor both sides

(y + 1)^2  =  -12(x + 2)          

-12(x + 2) = (y + 1)^2

And this is in the vertex form      4p(x - h) = (y - k)^2

Not in general....

For instance, let x = 3 and y = 4

Then

x^y  =  3^4  =  81

But

y^x  =  4^3  =  64 

1) 6x^2 + 2xy - 15x - 5y

factor by grouping

2x (3x + y ) - 5 (3x + y)        (3x + y)   is common to both terms

(3x + y)   (2x - 5 )

2) 2x^2 + 7x + 5    we're looking for two terms that multiply to 2 for the first term and to 5 for the last term......trial and error produces

(2x + 5) (x + 1)

Thanks for those pics, hectictar....I tried to explain what I did as well as possible, but I see how it can be difficult to follow....this one might be a good candidate for a "YouTube"  presentation....!!!!!

You really deserve more than a single point for all that effort...!!!

BTW.....I really liked this one....I still wonder how someone figured out that we would get a rational number as an answer....it seems, at first blush, that we would end up with some nasty radical....LOL!!!!

2ax - ay + 4bx - 2by

We can "factor by grouping"

a (2x - y)  +  2b (2x  - y)      note that  (2x - y)  is common

(2x - y) (a + 2b)

Wow....that's impressive, heureka!!!!....that might be one of the most difficult problems ever submitted to the forum.....!!!

It seemingly employed just about every facet of trig ( along with a few other things, too ) ...

Find the point with y-coordinate -28 on the line through (-62, 4213) with slope -69!

So we can solve this for the x coordinate of the point in the slope "formula"

[ 4213 - -28 ] / [ -62 - x] = -69    simplify

4241 / - [ 62 + x ]  =  - 69       multiply both sides by  - [ 62 + x ]

4241 = - 69  *  - [ 62 + x ]

4241 =  69  *   [ 62 + x ]

4241 =  4278 + 69x

-37 = 69x

-37/69   = x

So the point is (-37/69, -28)