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Q = (3,-1) and k = 30.

It's okay! Your solution was still really helpful. I figured the equation out, so it's okay! 

f(x) = x^4 - 24/5*x^3 + 16/5*x^2 + 26/5*x - 24/5, so f(1) = -1/5.

We have that angle TSU = angle QPS. Thus, we have the system:

8x-3=6x+11 = 2x=14 --> x=7.  

We can plug x back into the equation for angle TSU to obtain (8(7)-3)=(56-3)=\(\boxed{52}.\)

Final Answer: 52.

Some algebraic manipulation gets us an answer of -1. Double checked with wolframalpha as well.

There are 55 combinations which are all multiples of 4 as follows:

112 , 116 , 121 , 125 , 134 , 143 , 152 , 156 , 161 , 165 , 211 , 215 , 224 , 233 , 242 , 246 , 251 , 255 , 264 , 314 , 323 , 332 , 336 , 341 , 345 , 354 , 363 , 413 , 422 , 426 , 431 , 435 , 444 , 453 , 462 , 466 , 512 , 516 , 521 , 525 , 534 , 543 , 552 , 556 , 561 , 565 , 611 , 615 , 624 , 633 , 642 , 646 , 651 , 655 , 664 , Total =  55

Therefore, the probability is= 55 / 6^3 =55 / 216

since i haven't seen anyone answer yet, i'll try pitching my 2 cents.

we have that z+1/z = 1

if we square our given expression, we get:

z^2 + 2 + 1/z^2 = 1

z^2 + 1/z^2 = -1

if we square this again, we get:

z^4 + 2 + 1/z^4 = 1

z^4 + 1/z^4 = -1

as you might be able to deduce, this will equal -1 however many times we square our expression, because the 2ab term will always equal 2, and the RHS(right hand side) will always equal (-1)^2 = 1, so subtract the two, and you get -1.

z^2 + 1/z^2 = -1

z^4 + 1/z^4 = -1

z^8 + 1/z^8 = -1

Now we have a problem because we are asked to find z^10 + 1/z^10 which does not entail these powers of 2. We can try experimenting to get the z^10 term, and the first thing that comes to mind is multiplying by z^2 + 1/z^2

(z^2+1/z^2)(z^8+1/z^8) = z^10 + 1/z^10 + z^6 + 1/z^6 = 1

now we need the value of z^6 + 1/z^6 to get the value of our desired expression.

To do so, we can repeat these steps with z^2+1/z^2 and z^4 + 1/z^4

(z^2 + 1/z^2)(z^4+1/z^4) = z^6 + 1/z^6 + z^2 + 1/z^2 = 1

this is good, because we have the value of z^2 + 1/z^2 = -1

z^6 + 1/z^6 - 1 = 1

z^6 + 1/z^6 = 2

now we go back to our expression with the z^10 term and substitute the value of z^6 + 1/z^6 to get;

z^10 + 1/z^10 + 2 = 1

z^10 + 1/z^10 = -1

If you know how to count, you can do it.

Sorry, I can't see your diagram. It says it has been blocked by a moderator. 

Triangle Rule: The sum of the lengths of any two sides of a triangle must be greater than the third side.

1. Let's list all the sets. {8, 16, 21}{12, 17, 27}{8, 23, 31}{5, 8, 11}

2. Now is...

8+16 greater than 21?

12+17 greater than 27?

8+23 greater than 31?

5+8 greater than 11?

I'll leave that to you. 

By triangle inequality theorem, any two given sides, a, b, of a triangle, must be greater than or equal to the third side

a+b >= c

Just check all of the 4 cases for the answer, it will be the outlier that does not satsify the above condition. In other words, when you add two of the sides, it is smaller than the third side.

Sum of cubes factorization, which can be generalized to all odd powers of n:

a^3 + b^3 = (a+b)(a^2-ab+b^2)

replace a and b with c and 1/c respectively in this case to get:

c^3 + (1/c)^3 = (c+1/c)(c^2-1+1/c^2)

since we are already told c^2 + 1/c^2 = 14, we can substitute to get:

c^3 + (1/c)^3 = (c+1/c)(14-1) = (c+1/c)(13)

now all we need to find is the value of c+1/c

realize that:

(c+1/c)^2 = c^2 + 1/c^2 + 2 = 14 + 2 = 16 

this is because we are given that c^2 + 1/c^2 = 14, so just substitute

(c+1/c) = sqrt(16) = +-4

Since we want the largest value possible of the expression, we can assume it's positive, because if it's negative, then -4(13) would likewise be negative. Our answer is then:

4(13) = 52

We can solve this problem by assuming that all the side lengths are equal to 10. With the increase, all of them are 11. Finding the volume of both prisms gives us 1000 and 1331 respectively. Dividing the difference by the original gives us \(\frac{331}{1000}\) or 33% increase.

vieta's formulas would really help in this situation! With vietas, you don't need to find the specific roots a,b of the quadratic, but rather, you can derive and find the elementary symmetric sums of the roots (a + b + c, ab + bc + ac, abc as an example for a third degree poly.)

By vieta's, the roots a,b of the given quadratic are such that:

a+b = -4/2 = -2

ab = -5/2

Now that we have these two values, just expand the numerator as:

((a+b)-ab) = (-2+5/2) = 1/2 for the numerator

and rewrite the denominator as:
(a+b)^2 - 2ab = (-2)^2 +5 = 4 + 5 = 9

our answer is then:

(1/2)/9 = 1/18

First think of the inner f(x) as, say, y.

f(y) = 5    

so either y² - 4 = 5 or y + 3 = 5  

If y² - 4 = 5, then y = 3 is a possibility as 3 > -4, also y = -3 is a possibility as -3 > -4

If y + 3 = 5 then  y = 2, which is not possible as 2 is not less than -4

So there are two possibilities for y, which means you have to consider two possibilities for f(x)

(1)  f(x) = 3

      3 = x² - 4  means x² = 7  or x = ±√7  both of which are > -4

       3 = x + 3  means x = 0  which is not possible because 0 is not less than -4.

(2) f(x) = -3

     Can you do this one?

AB = 9.6    or  48 / 5     or     (47+1) / (6-1)  

1. 1/3 hr=4 1/2 km

If we want to get 1/3 hour to 1 hour, then we have to multiply by 3. Remember, if you do something to one side of the equation, you have to do it to the other side too for the equation to be true. Therefore, let's multiply 3 on both sides.

2. 3(1/3)hr=3(4 1/2)km=1hr=13 1/2 km

Answer: Nina can run 13 1/2 kilometers in one hour. 

AB = 47/6.

The area that the two triangles have in common is 8*sqrt(3).

Chris labels 4 points with the number 25.

wrong answer :(

Could i please have a hint, rather than the whole answer?

There are 37 students who take both languages.

The sum of all values of x is 2 + 12 = 14.