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1)

Compute the value of x such that \(\left(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}\cdots\right)\left(1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots\right)=1+\frac{1}{x}+\frac{1}{x^2}+\frac{1}{x^3}+\cdots.\)

2)

The graph of the parabola x = 2y^2 - 6y + 3 has an x-intercept (a,0) and two y-intercepts (0,b) and (0,c). Find a + b + c.

3)

Circles with centers of (2,2) and (17,10) are both tangent to the x-axis. What is the distance between the closest points of the two circles?

4)

Given the function y=x^2+10x+21, what is the least possible value of y?

5)

What real values of x are not in the domain of \(f(x)=\frac{1}{|x^2+3x-4|+|x^2+9x+20|}?\)

6)

Define the operation \(a\nabla b = b^a - 2\). What is the value of \((2\nabla 3) \nabla 2\)?

 May 5, 2020
 #1
avatar+652 
+3

Hey by the way next time please do one question per question!!! 

 

Thanks :D
 

coolsmileycool

 May 5, 2020
 #2
avatar+111455 
+2

6)

 

(2∇3)  = 3^2 - 2   =  7

 

(7)∇2   =  2^7 - 2   =  126

 

cool cool cool

 May 5, 2020
 #3
avatar+111455 
+2

5)   x^2  + 10x + 21

 

The  x coordinate  of the vertex  =  -10/[ 2 * 1 ] =  -5

 

The  y coordinate of the vertex is the min  =

 

(-5)^2 + 10(-5) + 21  =

 

-4

 

 

cool cool cool

 May 5, 2020
 #4
avatar+111455 
+2

3)

Circles with centers of (2,2) and (17,10) are both tangent to the x-axis. What is the distance between the closest points of the two circles?

 

The distance  between the  centers is

 

sqrt  [ (17-2)^2   + (10-2)^2]  = sqrt  [15^2  + 8^2]   =sqrt 289   = 17

 

The radius  of  the first circle is 2  and the radius  of   the second circle  is 10

 

So.....the  distance between the two closest points =  17 - 2 - 10    =  5

 

 

 

 

 

 

 

cool cool cool

 May 5, 2020

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