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wait, how did you get your last answer for question 2? :)

Thank you, CPhill

heureka

The hotter the heat source, the further away it can warm.

The sun is very hot so it impact is felt even further away than the Earth. 

Ten!  One to change the light bulb and nine to stand on each other's shoulders!

I usually just punch the number into the calculator to calculate. Maybe that's just me. (7.7 * 10¹⁶)

Which, without using a calculator, would be 77 followed by 15 zeros. "77 000 000 000 000 000"

9.1e18

=  9.1 * 10¹⁸

Multiplying any number by 10 moves the decimal point to the right one place.

For example.......     2 * 10  =  20

                                20 * 10  =  200

                                2.5 * 10  =  25

So, multiplying any number by 10, 18 times,  moves the decimal point to the right 18  places.

For this problem, we need to move the decimal point to the right 18 places.

=  9 100 000 000 000 000 000

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)

Yeah, I was thinking a video might be helpful..

I also wouldn't have expected a clean answer either...it beats me!!

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!!!!

I didn't understand at first so I tried doing it, I'll share some pictures in case it helps 

Since I didn't have any paper that was big enough to make it 12 inches by 9 inches,

I made it 12 centimeters by 9 centimeters.

Then, fold from the top right corner to the bottom left right corner ( I did it backwards..oops! ).

At that point, you can find what  x  is.

This picture shows why the side length is  x  and the hypotenuse is  12 - x  .

Then unfold it and draw a the line parallel to the bottom edge like this...

Now you're left with a right triangle where the legs are  9  and  12 - 2x  cm long, and the hypotenuse is the crease.  And looky here...it is right about 11 and a quarter centimeters long! 

I dont think I ever would have figured that one out on my own!!

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)

1.if a triangle is isosceles, then do its base angles HAVE to be congruent?

No....we just require that any two angles be equal

2.If a triangle has two congruent sides, does it have to be isosceles?

By the usual definition, if two sides are equal, the triangle is isosceles

3.If two angles of a triangle are congruent, does the triangle have to be isosceles'?

Yes....equal angles in a triangle mean that the sides opposite those angles are equal

4.Does an equilateral triangle have to be equilangular?

Yes.......equilateral triangles are always equiangular

5.Is the side opposite to the largest angle in a triangle the longest side?

Yep....the greatest angle is always opposite the greatest side

Solve for t:
log(2) = 0.00249688 t

log(2) = 0.00249688 t is equivalent to 0.00249688 t = log(2):
0.00249688 t = log(2)

Divide both sides by 0.00249688:
Answer: | t = 277.605

I did this with a sheet of paper [ pretending that it was 9  x 12 ] .......I orient the sheet with the smaller side at the top and fold from the top left corner to the bottom right corner

First.....fold as instructed......at the "bottom" of the sheet we will have a right triangle whose leg lengths are 9 and x  and whose hypotenuse is 12 - x

Using the Pythagorean Theorem to solve for x, we have

sqrt ( 9^2 + x^2)  = 12 - x     square both sides

9^2 + x^2  = x^2 - 24x + 144   subtract x^2 from both sides

81 = -24x + 144  rearrange

24x  = 63

x = 63/24  =  21 / 8

Now.....unfold the paper......if we draw a line from the bottom left fold of the crease  parallel to the bottom edge of the paper all the way across to the other side, we will have a right triangle whose sides are 9  and [12 - 2(21/8)] = [ 12 - 42/8] = [96 - 42] / 8 = ]54/8] =  27/4........and the crease will form the hypotenuse of this right triangle

So......using the Pythagorean Theorem again the length of the crease is given by

sqrt [  9^2  + (27/4)^2 ] =  sqrt [ 81 + 729/16] =

sqrt [ 1296 + 729] / 4 =  sqrt [ 2025]/ 4 =  45/4  inches = 11 + 1/4 inches

Since the parabola turns upward, we re looking for the  y value of a vertex that is  > - 4

So....we can solve the following to find the x coordinate of this vertex

-b/ (2 * a )  = x coord of the vertex

-b / (2 * 1)  = x

-b/2  = x

And  we require that

x^2 + bx + 12  > - 4       substituting, we have that

(-b/2)^2  + b (-b/2) + 12 > -4

b^2 /4  -b^2/2 > -16

-b^2 / 4  > -16

-b^2 > -64      multiply by -1 and reverse the  inequality sign

b^2 < 64

So

-8 < b  < 8  will produce a parabola whose range > - 4

And.....the graph is still in range whenever the largest integer value of b  = 7

For comparative purposes......see the graphs here for some different values of "b"

https://www.desmos.com/calculator/my77qzagdu

Note that  when   l b l < 8  the range is > -4

But when  l b l ≥ 8,  the graph is out of range

x + ( - .50 )  =  4

x  -  0.5  =  4                 Add  0.5  to both sides of the equation.

x  =  4.5

x  =  45/10                    Reduce the fraction by 5 .

x  =  9/2

11y=-42

y=-42/11

This graph should help (note: for x>=1 the two are the same):

At first glance, these two look the same -- after all, one could simplify  \(\sqrt{\frac{x+1}{x-1}}\)into \(\frac{\sqrt{x+1}}{\sqrt{x-1}}\).

However, if we try to graph these using an x-y table:

As you can see, \(\sqrt{\frac{x+1}{x-1}}\)allows for negative values, but \(\frac{\sqrt{x+1}}{\sqrt{x-1}}\)doesn't.

My advice: when in doubt, plug in a few numbers of graph it. 

This is the well known example of the 3-4-5 right angled triangle (each side is just twice 3, 4 and 5 respectively). So, yes, it must be a right angled triangle.

You can check using Pythagoras theorem: 6^2 + 8^2 → 36 + 64 → 100 and 10^2 → 100

i.e. The square on the hypotenuse is equal to the sum of the squares on the two adjacent sides (only true for right angled triangles).

Oops!  See CPhill's reply.

Part 1.  \(18\sqrt8\rightarrow 18\sqrt{2^2\times2}\rightarrow18\times2\sqrt2\rightarrow36\sqrt2\)

Part 2. \(8\sqrt{18}\rightarrow8\sqrt{3^2\times2}\rightarrow8\times3\sqrt2\rightarrow24\sqrt2\)

Part 3.  Now you have parts 1 and 2 you should be able to have a good go at this yourself.

After they pass at 9PM, they continue traveling for 1.25 hours

So...let the rate of the slower cyclist = R and the faster R + 4

And rate * time  =  total distance

So.....

Distance traveled by slower cyclist after passing + Distance traveled by faster cyclist after passing = 40 km

So....

R * 1.25 + (R + 4) * 1.25  =  40   simplify

1.25R + 1.25R + 5 = 40  

2.50R  + 5  =  40     subtract 5 from both sides

2.50R = 35      divide both sides by 2.50

R = 14 k/ hr    =  rate of slower cyclist

And R + 4  = 18 km/ hr  =  rate of faster cyclist