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The parabolas defined by the equations y = -x^2 - x + 3 and y = 2x^2 - 1 intersect at points (a, b) and (c, d), where c is greater than or equal to a. What is c - a? Express your answer as a common fraction.

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\(y = -x ^ 2 - x + 3\\ y = 2x ^ 2 - 1\\ \color{black}2x^2-1=-x^2-x+3 \)

\(3x^2+x-4=0\)

\(x = {-1 \pm \sqrt{1+4\cdot 3\cdot 4} \over 2\cdot 3}\\ x=\frac{-1\pm 7}{6}\)

\(a=-\frac{4}{3}\\ \color{black}b=\frac{23}{9}=2.5\overline {5}\\ c=1\\ d=1\)

\(c-a=1-(-\frac{4}{3})\)

\(c-a=\frac{7}{3}\)

  !

- x^2 -x + 3    = 2x^2 -1      re-arrange

0 = 3x^2 +x-4             use quadratic formula to find the x values where the graphs intersect

                                           a = 3     b   = 1      c = -4

\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

                                          use the x values in the equations to calculate the corresponding 'y' values

(actually....you will only need the 'x' values  which will be   c    and a  )

                                                ...then you can answer the rest of the question

Missing info .....  

"The elevation of city D is ??  of the elevation of city C."

.25   +   .30    +  .75     -         .20        =    amount in the jar

You have 6 letters: A, B, C, D, E, F. Since they can be repeated, you will therefore have: 6^4 =1,296 permutations, or 4-letter "words". But each "word" must have at least one consonant. Therefore, we have to remove ALL 4-letter "words" that are only vowels. There are very few such permutations. Beginning with "A", there are only 8 such permutation as follows:

AAAA, AAAE, AAEA, AAEE, AEAA, AEAE, AEEA, AEEE.
Similarly, beginning with "E", there are also 8 such permutations.

So then, the total number of 4-letter permutations or "words" =6^4 =1,296 - 8 - 8 =1,280 such 4-letter "words"

The proportion of cups of chocolate chips to cookies is constant and can be simplified as .Thus, if  is the number of cups of chocolate chips it will take to make  cookies, . Solving for x gives an answer of 10 cups.

😁😁😁

Apologies, I've misled you by reacting too quickly, with too little thought!  

With dot products  u•v = |u|*|v|*cos(theta), where |u| is the magnitude of u, and theta is the angle between u and v. This means that

v•u = |v|*|u|*cos(theta) = u•v 

The same is true for the other dot products.

34xy/x    -47      the x's will 'cancel each other'

34y  - 47

Problem 1. One digit of 970405 can be changed to make the result divisible by 225. What is the six-digit number after the modification?

970425 / 225 = 4313 

(2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47) >>Total = 15 such divisors

9 , 11 , 13 , 15 , 17 , 19 , 23 , 25 , 27 , 29 , 31 , 33 , 37 , 39 , 41 , 43 , 45 , 47 , 51 , 53 , 55 , 57 , 59>Total =23 such numbers

Relatively Prime means that the GCD of any of the above 23 numbers and 14==1

80 out of 360  of the total area of the circle is 48 pi

80/360 pi r^2 = 48 pi

r^2 = 48 (360/80)                   I think you can finish !

There are 127 integers in base 7 to use the digit 4 in the first 343 integers without counting the repeats such as 44 or 444.....etc. If you include the repeats, then there are 147 integers that use the digit 4.

It would be

3(v•u) - 2(w•u) = -3(u•v) + 2(u•w) = -3x3 + 2x4 = -9 + 8 = -1

right?

Use quadratic formula

\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)            a = 2     b = -6     c = 7       to find the   m +- ni    solutions, then find  m*n 

180=abs((30*2) - (5.5*M)), solve for M

M = 480 / 11 =43.636363.....

Time =2:43:38

The largest is   6   out of ( 1+2+3+6)   =    6 / 12   of the entire 360^o of the entire circle ...

    6/12 x 360 = 180 degrees

n = 1,  7,  21,  49

You're welcome ! 

sumfor(n, 1, 1000, (n/2^n))=It converges to 2

Partial sum formula =2^(-k) * (-k + 2^(k+1) - 2)

Use substitution to find the values of "a" and "b":

a = 30   and   b= 14

30^2  -  14^2 =704    and    30  +  14 =44

Hour hand starts at 60 degrees and moves 1/2 degree per minute  1/120 degree per sec

  minute hand moves 1/10 degree per second

60 + 1/120 x    =   1/10 x  -180  

240 + 1/120 x = 1/10 x 

240 = 1/10 x - 1/120 x

240 =  12/120 x = 1/120 x

240 = 11/120 x

x = 2618.18 seconds after 2 o'clock

minute hand will be at   43.63 minutes  =  43:38

  ~ 2 : 43 : 38