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The triangle inequality theorem states that the sum of any two side lengths of a triangle must be greater than the third side length. So we have the following three inequalities:

AB+BC>AC

AC+BC>AB

AB+AC>BC

Substituting in the expressions for the side lengths, we get the following three inequalities:

3x−3+x+7>x+15

x+15+x+7>3x−3

3x−3+x+15>x+7

Simplifying these inequalities, we get:

4x+4>x+15

2x+22>3x−3

4x+12>x+7

Solving for x in each inequality, we get:

x>11

x>−19

x>5

The largest possible value of x is 11, since the side lengths of a triangle must be positive. The smallest possible value of x is 5, since the third inequality above is an equality when x=5. Therefore, the range of all possible values of x is [5,11]​.

1 - Start at the top: sqrt((sqrt(3) / 2 + 1)/2)=0.9659258262890682867497431997289
2 - [1 - 0.9659258262890682867497431997289]
3 - sqrt[0.0340741737109317132502568002711 / 2]
4 -sqrt [0.13052619222005159154840622789549+1]/2
5 - sqrt[0.56526309611002579577420311394775]
6 - [(0.75183980747897739640751940637696)+1] / 2
7 - sqrt[0.87591990373948869820375970318848]
8 - 0.93590592675732570029170724946674
9 - Arccos[0.93590592675732570029170724946674]

10 - [11 Pi] / 96  [in radians]
Note: This question is slightly different from the above link, which gives: [13 Pi] / 96.

The system of equations can be written as

6x^2 - 9x + c - x^2 - x = 5x^2 - 3x 5x^2 - 10x + c = 0

The discriminant of this quadratic equation is

b^2 - 4ac = 100 - 4(5)(c) = 100 - 20c

(a) The system has exactly one real solution when the discriminant is 0.

100 - 20c = 0 => c = 5

(b) The system has more than one real solution when the discriminant is positive.

100 - 20c > 0 => c < 5

(c) The system has no real solution when the discriminant is negative.

100 - 20c < 0 => c > 5

Therefore, the system has:

Exactly one real solution when c = 5.

More than one real solution when 0 < c < 5.

No real solution when c > 5.

Here is a table summarizing the results:

c | Number of real solutions

---|---

0 < c < 5 | More than one

c = 5 | Exactly one

c > 5 | None

Hope this helps!

The answer is 13*pi/96.  The solution has been posted here:

https://web2.0calc.com/questions/pre-calc-problem

In order for 23x2+kx−5 to factor as a product of linear binomials with integer coefficients, the two linear terms must have a common factor of 23. This means that k must be a multiple of 23. The only multiples of 23 that work are k=−23 and k=0, so the only possible values of k are −23,0​.

Here is the proof of the claim that the two linear terms must have a common factor of 23. Suppose that 23x2+kx−5 can be factored as a product of linear binomials with integer coefficients:

23x2+kx−5=(ax+b)(cx+d)

for some integers a, b, c, and d. Then both ax+b and cx+d must be divisible by 23, since the only common factor of 23x2 and −5 is 1. But this means that k=ad+bc must also be divisible by 23.

Correct! xD

According to Google it has no sides BUT

I am quite sure it has an inside and an outside.    

Guest answer #3 is correct.    Good work  

Guest answer #1 is almost correct, They just forgot to include combinations with 0

Here is my answer.

D==Dean's money,  P==Patrick's money

d==How much Dean spent,  p==How much Patrick spent

D/P == 3/8..................... (1)  

d/p==3/5........................ (2)  

D - d ==1/4D.................. (3) 

P - p==136..................... (4), solve for D, P, d, p

From (1) and (2) above, we have:

d==0.60p   and  D==0.375P        (5)

Sub d  and  D in (3) above and we have:

0.375P - 0.60p ==1/4 * 0.375P

P - p == 136, solve for P, p 

P==$256 - Patrick's money

p==$120 - what Patrick spent

Sub P and p into (5) above:

D==0.375 * 256==$96 Deans money

d==0.60 * 120==$72 - what Dean spent.

We can simplify the expression for an​ as follows:

a_n = 1 + sqrt(1/n) - sqrt(1/(n + 1)) - sqrt(1/n - 1/(n + 1)) = 1 + sqrt(1/n) - sqrt(1/n + 1/(n(n + 1))) - sqrt(1/n - 1/(n(n + 1))) = 1 + sqrt(1/n) - sqrt((n + 1)/(n(n + 1))) - sqrt((n - 1)/(n(n + 1))) = 1 + sqrt(1/n) - (n + 1)/n - (n - 1)/(n + 1) = (n + 1)/(n + 1) - (n + 1)/n - (n - 1)/(n + 1) = 1 - 1/n

Then the product a_1 a_2 a_3 works out to 1/99.  The answer is (C).

Correct, troll. 

n is 5 because 5 * 5! +5! = 720. I am not sure what *** means, but I think it means going on until something, which is actually true. 5! = 5 * 4 * 3 * 2 * 1. You can also search it up. I'm not sure why they put *** though. It is more clear to put ......

You do not need to answer then, guest. 

Nice work. 

Correct!

Correct!