Solve the equation below. List all the solutions. Include all complex solutions.
4x^2−8x+15=0

Step-by-step explanation:

A=4, B=-8, C=15

the discriminant B^2 - 4ac = (-8)^2 - 4(4)(15)

                                  = 64 - 16(15)

                                 = 64 - 240

                                = -176

                           = -1 * 11 * 4 * 4

Since this discriminant is negative, there are two imaginary/complex solutions.

the square root of this discriminant is 4 i sqrt(11)

the two solutions are  [8 +or- 4 i sqrt(11)]/8

                                  = [2 +or- i sqrt(11)]/2

check:

it will suffice to show 4x^2 - 8x = -15        

x^2 = [2 + i sqrt(11)]/4 =  [4 + 4 i sqrt(11)  - 11]/4

                       = [4 i sqrt(11) - 7]/4

then 4x^2 = 4 i sqrt(11) - 7  <--- 4 cancels

and 8x = 4(2 + i sqrt(11) ) <-- 8 cancels 2

        = 8 + 4 i sqrt(11)

So 4x^2- 8x = 4 i sqrt(11) - 7  - [ 8 + 4 i sqrt(11)]

                    = 4 i sqrt(11) - 7 - 8 - 4 i sqrt(11)

                    = -15

which is sufficient to show that the answer is correct.

the two solutions are  [2 +or- i sqrt(11)]/2

are the official answers

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.

the first one, all those points on the line just make a strait line.