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To find the x-intercepts:         y  =  x² + 4x - 12

                                              0  =  x² + 4x - 12

                                              0  =  (x + 6)(x - 2)

x-intercepts:                                   x = -6  and  x = 2

To find the vertex:                 y  =  x² + 4x - 12

                                      y + 12  =  x² + 4x

                                y + 12 + 4  =  x² + 4x + 4

                                       y + 16  =  (x + 2)²

vertex:                                   (-2, -16)

The endpoints of the base of the isosceles triangle are (-6, 0) and (2, 0)     --->     so its base is 8

The height of the triangle reaches from the midpoint of the base (-2, 0) and the vertex (-2, -16)     --->     so its height is 16

The area is  ½ · base · height  =  ½ · 8 · 16  =   64

If you divide this by using long division, you will get a quotient of 3x² + 4x + 5 and a remainder of (a + 20)x + (b + 25).

Therefore a = -20  and  b  =  -25.

The appropriate formula is  n(n - 3)/2  where n is the number of sides:

--->     7(7 - 3)/2  =  7(4)/2  =  14

This is a binomial distribution: one formula for solving this problem is:

  ₁₅C₁₁·(.86)¹¹·(.14)⁴ + ₁₅C₁₀·(.86)¹⁰·(.14)⁵ + ₁₅C₉·(.86)⁹·(.14)⁶  + ... + ₁₅C₁·(.86)¹·(.14)¹⁴ + ₁₅C₀·(.86)⁰·(.14)¹⁵

I'll let you calculate this (there is a simpler way).

The parabola that has a minimum value at (2, 10)  is:  y - 10  =  2(x - 2)²

Multiplying this out:                                                       y - 10  =  2(x² - 4x + 4)

                                                                                      y - 10  =  2x² - 8x + 8

                                                                                            y  =  2x² - 8x + 18

So:  c = 18

If you add down the first column you will get:  (n + 2) + (3) + n - 2)  =  2n + 3

If you add down the third column you will get:  (n) + (2n - 5) + (2)  =  3n - 3

Being a magic square, they must be equal:  2n + 3  =  3n - 3

                                                                              6  =  n

Let's look at the worst case scenario  ....the first two people you meet are 1 Canadian and 1 Norwegian.....the NEXT person you meet will be one of those two.....and you will have met two people of the same nationality   soooooo   3

8.8  inches/ sec   x   60sec /min   x  60 min/hr  x  1ft/12 inches x 1 mile/5280 ft  =

8.8 x 60 x 60 / (12 * 5280)  =        m/hr

Note how everything cancels out except  miles  and    / hr

actually nvm but why do you have to multiply it by 7 and also why did you need to multiply these two together 7C2*5C2?

You can also do this without making a common demoninator. If the numbers were increased to i.e something like 15234/15235 and 15676/15677, finding the common denominator would be reduntant and be extremely time confusing. Instead, we can notice that both the fractions are 1/n away from 1. 1/5 is clearly greater than 1/7, as spliting the same thing into 5 things compared to 7 things, is clearly greater. Thus, we are subtracting 1/5 from 4/5, SOMETHING THAT is greater than subtracting 1/7, so therefore 6/7 is greater .

Find the rate PER MINUTE

  250 cans/hr   x   1 hr/60min  = 4.1666 can/min

4.1666 can/min  x 25 min = 104 cans

Or use cans per hour....but then you'll need to convert 25 min to hours   25/60 = .4166 hr    etc

 $5.60/14 pen   x 3 pen   =  $1.20        See how things 'cancel out' in these solutions ....if you set them up correctly ! 

There are LOTS of ways to write  equaivalent expressions.....is this   5^4x-3      ?

     =   5^4x / 5³

       = 5^4x /125      are a couple equivalent expressions  

Take 1/2 of the middle    16x       1/2 is 8    square it 64

    if c is 64 then

(x+8)^2

i'm still confused why did you have to multiply 7*6 together, 7*6C2 etc and specifically why those numbers?

nCr is a little formula, (which most times you do not even have to know, the formula I mean)

Anyway

nCr  which is available on just about any calculator means

how many ways can r objects be chosen from n objects.

So if you have 10 people and you want to know how many different groups of three you can make that would be 10C3 ways.

----------------

Now look at the first question i did.

(a) While travelling abroad, I bought 7 identical bags of candy for 3 friends. How many ways can I distribute the candy to my 3 friends, so that each friend gets at least one bag of candy?

7 identical bags shared between 3 friends but they all get at least one each.

So really it is how many ways can 4 bags be given to three different people.

Here are the 4 bags      ****

I need to break them up into 3 groups. I will use bars to seperate them. I need 2 bars.

Person 1 gets the bags in front of the first bar

Person 3 gets the bags behind the last bar

Person 2 gets the bags in the middle.

eg

||****   

This would be one possibility.  Person 1 and 2 get none and person 3 gets all four of the bags.

The bars can go anywhere.

So  the number of possibilities is     6 items, choose 2.  Those two will be the bars, (people)

6C2 ways.

this is called the stars and bars method by the way.

If they cut 48 sections then that means that the original number of sections is times 0.4 (40%)
This can be expressed as x*0.4=48

Solving for x, simply rearrange to x=48/0.4
48*04=120 => x=120

You can double check this by 120*0.4 which is 48.

Thank you!!

how does the c(n, r) work, like how but mainly why do you have to do c(n, r) with those specific numbers and where did you get those numbers?

Hello ElectricPavlov,

what is the tallest similar chimney that can be painted with 8 litres of paint?​

\(Area \sim (distance) ^ 2\\\) 

Greetings   !

1) There are 18*17*16 ways to choose the three seats, if people can be adjacent.  We must subtract the cases where there are two people adjacent, which is 18*13*6.  So the number of ways is 18*17*16 - 18*13*6 = 3492.

2) Anty can get to any point inside the square with vertices (5,5), (5,-5), (-5,-5), (-5,5), so there are are a total of 121 points.

We can place one of the unique beads, and call it reference number 1.  Then the possible places of the two same colored beads are at 2 and 6, 3 or 5 and 4, so three ways.  The rest of the 4 charms can be placed in 4! ways.  So the number of arrangements is 3*4! = 72.

This forms a right triangle with hypotenuse = 13 ft

   5 is one leg   the other leg is  L₂

Use pythagorean theorem

5^2 + L₂ ²  = 13²      Where L₂ is the other leg of the right triangle....solve for L₂

Draw a picture....     hypotenuse = 100 ft

angle = 65 degrees

sin 65 = opp/hyp = height/100

100 sin 65   = height