12 divided by the sum of twice a number and 3 is the same as the number

Let the number be x.

Then -12 / (2x+3) = x

Remove fractions by mult.  each term by (2x+3):

-12 = x(2x+3), or 2x^2 + 3x + 12 = 0

Use the quadratic formula to solve this.  

a=2, b=3, c=12
                 -3 plus or minus sqrt (3^2 - 4(2)(12)
Then x =
                                       4

Unfortunately, the discriminant   9 - 48 is negative, so this equation has only complex roots, no real roots.

Please double-check to ensure that you have shared this problem exactly as it was originally printed.

Let me make one change in the problem and then determine whether or not this problem is solvable:

+12 divided by the sum of twice a number and 3 is the same as the number.

   12                  x
=  Cross multiplying, 12 = x(2x+3), or 2x^2 + 3x - 12 = 0
 2x + 3              1
                                                                          -3 plus or minus sqrt(102)
Applying the quadratic formula results in  x =
                                                                                             4

Let's go back to "-12 divided by the sum of twice a number and 3 is the same as the number."

Then 

  -12        x
= ---
2x+3        1

Cross multiply:  -12= (2x+3)x

Sorry, but my efforts here are going nowhere.  Hope they give you some guidance in what to do yourself.  Again, please ensure that you have copied down this problem exactly as it was presented in the original form.

Answer: 10pi

x = central angle apc = measure of minor arc ac

the two angles x and 30 add to 180, so x+30 = 180 and x = 180-30 = 150. the measure of minor arc ac is 150 degrees.

what we need to do from here is find the circumference and then multiply it by x/360 since there are 360 degrees in a circle. in this case, x = 150, so x/360 = 150/360 = 15/36 = 5/12. the shortest distance along the curve from a to c is 5/12 of the distance around the entire circle (aka circumference)

c = circumference

c = pi*d

c = pi*24

c = 24pi

now multiply this by 5/12 to get

distance along minor arc ac = (5/12)*(circumference)

distance along minor arc ac = (5/12)*(24pi)

distance along minor arc ac = 10pi

the formula you can use is

l = (x/360)*pi*d

with

l = length of minor arc

x = central angle

d = diameter

which essentially is the steps we took though a bit differently.