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Oops!

Electric Pavlov found that I switch the 115 to only 15; so the equation should be:  y  =  (115 -3x) / (1 + x).

However, the point remains the same.

You really helped me understand it! Thanks :)

I don't understand how they can intersect.

Their centers are on a vertical line 9 units apart (from -1 to 8).

The first circle has a radius of 4 while the second circle has a radius less than 4.

How can they intersect?

Multiple ways to solve this

  rememebr   tan = sin / cos       you are given cos y      this is also  sin x

           find cos x          via     cos ^2 + sin^2 = 1

                                                 cos^2 = 1 - sin^2

                                                  cos^2 = 1 - (7/29)^2 = 1 - 49/841 = 792/841

                                                    cos x = sqrt (792)/29 =   6 sqrt 22 /29

tan x = sin x / cos x =  6 sqrt 22 / 29   /  7/29 =  6 sqrt 22 / 7

Remember in a rt triangle   sin = opp side / hypotenuse

we will need to find side DC 

Pythag theorem    DC = sqrt (9^2 - 5^2) = sqrt 56

sin b =   sqrt 56  /  9 = 2 sqrt 14 / 9       = .83148   

......or you can find cos b = 5/9     then find   sinb   with the trig identity    cos^2 + sin^2 = 1

                                                                                                                      25/81  + sin^2 = 1

                                                                                                                          sin^2 = 1- 25/81

                                                                                                                                   = 56/81

                                                                                                                            sin = sqrt 56 / 9 =  2 sqrt 14 / 9        as before.....

Here is a desmos picture:

AB^2 = 22.

Here is a second way:

the two given points are equidistant from the center    (x, 0)

Use distance formulas for the two points and the center

(x-0)^2  + 4^2 = (x-1)^2 + 8^2

   solve for x = 49/2      so circle center is   49/2 , 0

use one pont to solve for r in the circle formula   (0,4 )

( 0 - 49/2)^2 + (4-0)^2 = r^2

   r^2 = 616.25                       r = sqrt (616.25)

The center of the circle is on the perpendicular bisector of each chord of the circle.

The plan:

1)  find the equation of the perpendicular bisector of the chord whose endpoints are  (0, 4)  and  (1, 8),

2)  find where this equation crosses the x-axis; this will be the center.

3)  find the distance from this point (the center) to the point  (0, 4).

1)  The midpoint of  (0, 4)  and  (1, 8)  is  ( ½, 6).

      The slope of the chord is:  m  =  (8 - 4) / (1 - 0)  =  4.

      The perpendicular will have a slope of  - ¼.

      The equation of this line is:  y - 6  =  - ¼( x - ½ ).

      Multiplying by -4:     -4y + 24  =  x - ½

      Multiplying by 2:       -8y  + 48  =  2x - 1

2)  On the x-axis, the value of y is zero.

     Making this subsitution:  -8(0) + 48  =  2x - 1

                                                        49  =  2x     --->     x  =  24.5

      So, the center is  (24.5, 0).

3)  Finding the distance from  (24.5, 0) to (0, 4):

     distance  =  sqrt(  (4 - 0)²  +  (0 - 24.5)²  )     =     sqrt( 16 + 600.25  )     =     sqrt(  616.25  )

r = sqrt(6497) / 16

Using the point-slope form to find the equation of the line through (7,8) and (-9,0):

The slope is:  ( 0 - 8 ) / ( -9 - 7)  =  -8 / -16  =  1/2

The equation is:  y - 0  =  (1/2)( x - -9 )     --->     y  =  (1/2)(x + 9)     --->     y  =  0.5x + 4.5

Combining the two equations:    2x + 10  =  0.5x + 4.5

                                                         1.5x  =  -5.5

                                                              x  =  -11/3

Using this value to find y:  y  =  2( -11/3 ) + 10     --->     y  =  8/3

Add these two value together to get your answer.

You have probely done it by now

*** edited ***   Equate the two parabolas and solve for x to find the x coordinates of where they intersect

x^2 + 4x + 6     =    1/2 x^2 + x + 4

1/2 x^2 + 3x + 2 = 0        use quadratic formula to find    x=−3 +- sqrt5   <===== this is a and c

    you should be able to finish this......   

Using the work of Electric Pavlov:

If you leave the solutions in radical form, the answer is exactly 14.

Solutions:  [ 18 + sqrt(188) ] / 4     --->     [ 18 + 2·sqrt(47) ] 4     --->     [ 9 + sqrt(47) ] / 2

and            [ 18 + sqrt(188) ] / 4     --->     [ 18 - 2·sqrt(47) ] 4     --->      [ 9 - sqrt(47) ] / 2 

If  a  =  [ 9 + sqrt(47) ] / 2    then    a - 1  =  [ 7 + sqrt(47) ] / 2    and    1 / (a - 1)  =  2 / [ 7 + sqrt(47) ]

If  b  =  [ 9 - sqrt(47) ] / 2    then    b - 1  =  [ 7 - sqrt(47) ] / 2    and    1 / (b - 1)  =  2 / [ 7 - sqrt(47) ]

Multiply both of these fractions by the complement of their divisors to get:

2[ 7 - sqrt(47) ] / [ 49 - 47 ]     +     2[ 7 + sqrt(47) ] / [ 49 - 47 ] 

=     [  14 - 2·sqrt(47) ]  +  [  14 - 2·sqrt(47) ]      /  2

=             28 / 2

=                14

James the fly starts at (0, 0) Each step, James moves one unit right or one unit up. He is trying to get to the point (5, 7). However, at (4, 3) there is a frog that will eat him if he goes through that point. In how many ways can James reach (5, 7)?

First I am going to consider how many ways assuming there is no fly eating frog

Each step must be up or right.  There are 5 steps to the left and 7 steps up.

so there are   12C5    or  12C7 ways = 792ways

Now how many of these will be unfinished because the frog will get its supper?

that would be 7C3* 5C1 = 35*5 = 105 paths for the frog to get its supper.

So there are   792 - 105 = 687 safe routes.

Let A and B be real numbers such that
\(\dfrac{A}{x-5}+B*(x+1)=\dfrac{-3x^2+12x+22}{x-5}\).
What is A+B?

\(\small{ \begin{array}{|lrcll|} \hline & \dfrac{A}{x-5}+B*(x+1) &=&\dfrac{-3x^2+12x+22}{x-5} \quad | \quad *(x-5) \\\\ & \mathbf{ A+B*(x+1)(x-5) } &=& \mathbf{ -3x^2+12x+22 } \\ \hline 1.~ x=-1:& A + 0 &=& -3*(-1)^2 + 12*(-1) + 22 \\\\ & A &=& -3-12+22 \\ & \mathbf{A} &=& \mathbf{ 7 } \\ \hline 2.~ x =-2:& 7+B*(-1)(-7) &=& -3*(-2)^2 + 12*(-2) + 22 \\ & 7+7B &=& -12-24+22 \\ & 7B &=& -14-7 \\ & 7B &=& -21 \quad | \quad : 7 \\ & \mathbf{B} &=& \mathbf{ -3 } \\ \hline & A+B &=& 7-3 \\ & \mathbf{ A+B } &=& \mathbf{ 4 } \\ \hline \end{array} }\)

I doubt that is 'obvious', or even makes any sense to over 98% of the people who use this forum.

So what?

It is a very old question, did you want an answer for some reason?

Hi AskNHelp,

I know what a Capybara is.  I have seen them in the wild in Peru.   

So I knew your icon was a Capybara.   :)

Mine is an Australian road sign.  Its message to motorists is fairly self-evident.   

Hey, Melody, I am a fan and I am very thankful that you unflagged and responded to my question. :)

Thank you and MathProblemSolver101 for answering. 

I didn't know how to do some of the hard problems, so I thought of posting it here, but I saw that the problems were copyrighted, so I decided to ask. I am lucky I asked xD. Thanks for responding, and about the thing about checking the agreement I made, I couldn't find it, sorry. 

Thank you.

Btw you should search up what a capybara is if you don't know what it is.