Pangolin14

1, 9, 36

What determines how many roots an equation has?

Its the... DISCRIMINANT, also known as b^2-4ac!

If the discriminant < 0, there are no real solutions. If the discriminant = 0, there is 1 real solution! However, if it is >0, there are two real solutions, as desired! Therefore, we have a=2 and c=6, so b^2-48>0, or b^2>48, so $b>\sqrt{48}.$

By vertical angles, w=56˚. 

Draw a line parallel to each of the vertical lines that passes through the middle vertex.

Then

So y=180-68-56=x.

so w=56

y=x=56

b=68

Pangolin14

Hello Guest, 

Please note that CPhill did make a mistake, and I didn't understand it either.

Thus I provide a new solution for this problem:

Here is my image at this link:

Favorable/Possible

Possible=6C2=15

Favorable= 8:

1,2

1,3

1,5

1,6

4,2

4,3

4,5

4,6

8/15

ok

6x-8=-48

12x-4=-48

12+-4=8

(6x+12)(x-4)

x=-2

x=4

The planetarium sells movie tickets to adults and students. Marsha bought 2 adult tickets and 3 student tickets for $27.50.$ Jay bought 1 adult ticket and 2 student tickets for $16.25.$

A. Write a system of equations to represent the situation.

Let $a$ and $s$ be variables representing the cost for each of the adult tickets and student tickets, respectively. Then the equation representing the total cost for Marsha's tickets is $2 a + 3 s = 27.50.$ 2a is the total cost for adult, and 3s is the total cost for student tickets. Now we will deal with Jay's tickets. The equation for Jay's total cost would be $a+2s=16.25.$ For the same reasons, a and 2s are total costs for Jay's adult and student tickets. The system of two fully simplified equations and two variables required to solve this question are:

$$2 a + 3 s = 27.50.$$

and

$$a+2s=16.25.$$

B.  Determine the cost of each adult ticket and each student ticket. (Show your work or explain how you used a graph to determine your answer.)

We know how to solve a system of equations, either by substitution, or elimination. Elimination seems to be the most efficient, as $2 \cdot a = 2a,$ and to our convenience, $2\cdot 2s-3s=s,$ which allows us to solve our equation by multiplying $a+2s=16.25$ by two and subtracting $2 a + 3 s = 27.50.$ That yields $s=5.00,$ and thus $a=6.25.$ 

C. Marsha bought additional tickets for her friends that decided to come to the movie. She bought a total of 4 adult tickets and 7 student tickets. What was her total cost? (Show your work.)

From #B, we know adult tickets ($a$) cost 6.25 and student tickets ($s$) cost 5.00, so we can multiply 6.25 per 4 tickets for 25 and 7 times 5 for 35. So her total cost is 25+35=$60 

Hello Noori,

To solve this problem we need to draw a diagram. We know that the tangents will be perpendicular to the radii drawn from their points of tangency, I just proved that yesterday, so we can draw a diagram, assume A' and C' are the centers of the tangent circles. Also, A, B, and C are colinear. Try to find the smallest distance between these lines, and build on this diagram!

Sorry, that's wrong!