Find an integer c such that the equation 4x^3 + cx -27 = 0 has a double root?

I really don't get this equation and I could really use some help. Anyone think they can help me out?

Find an integer c such that the equation 4x^3 + cx -27 = 0 has a double root?
if the roots are equal:


c^2-4*4*(-27)=0


or, c^2+4*4*27=0


or, c^2=(-4*4*27)


or, c=sqrt(-4*4*27)=4*3*sqrt(-3)


or, c=12*(sqrt 3)*i which is the answer.


I want to remind you that it is absolutely impossible to find the exact value of c since x is also unknown.To determine one quantity in quadartic equation with two unknown values is impossible.I think you got it?
Reply:4x^3 + cx - 27 = 0





(x-a)(x-a)(x-b) = 0


(x² - 2ax + a²)(x - b) = 0


x³ - bx² - 2ax² + 2abx + a²x - a²b = 0





Multiply of 4 to match the coefficients of x³


4x³ - 4bx² - 8ax² + 8abx + 4a²x - 4a²b = 0





Compare the coefficients of the powers of x


x²: -4b - 8a = 0


4b = -8a


b = -2a





x: 8ab + 4a² = c


-16a² + 4a² = c


c = -12a²





x^0: -4a²b = -27


8a³ = -27


(2a)³ = -3³


2a = -3


a = -3/2


c = -12(-3/2)²


c = -12 * 9/4


= -27
Reply:(2x - a)²(x - b) = 4x³ + 0x² + cx - 27





4x³ - 4bx² - 4ax² + a²x - a²b = 4x³ + 0x² + cx - 27





-4b - 4a = 0


b = -a





-a²b = -27


a²b = 27


a²(-a) = 27


a³ = -27


a = -3





b = -(-3) = 3





(2x - a)²(x - b) =


(2x + 3)²(x - 3) =


4x³ -27x - 27 = 0





c = -27





4x³ -27x - 27 = 0


Has a root at 3 and a double root at -3/2
Reply:(x-a)^2 (4x+b) = 4x^3 + cx - 27


(x^2 - 2ax + a^2)(4x+b) = 4x^3 +cx - 27





................ x^2 - 2ax + a^2


X... ....... ...........4x+b


____________________________


........... .... b x^2 - 2a b x + a^2 b


...4x^3 - 8 a x^2 + 4 a^2


_____________________________


4x^3+ (b-8a)x^2+(4a^2 - 2ab)x + a^2 b





a^2 b = - 27


b-8a = 0


4a^2 - 2ab = c





b = 8a





Hence a^2 * 8a = - 27





a^3 = -27/8





a = -3/2


b = 8a = -12





Since 4x - 12 = 4(x-3), then 3 is a root.





And





4a^2 - 2ab = c





c = 4*(-3/2)^2 - 2(-3/2)(-12)





c = 9 - 36





c = -27





Let's now check these results





............ 4...........0............ -27......... -27


-3.................... -12 ......... 36... ...... .27


............ 4........ -12............ 9............ 0


-3/2 .................. 6 ......... ..-9


............ 4......... -6.............. 0


-3/2 .............. .. 6


........... 4 ......... 0





Nice exercise





Ana

marguerite